OP Malhotra Class 12 Maths Solutions Chapter 26 Application of Calculus in Commerce and Economics Exercise 26 (A)

Question 1.
(i) A manufacturer of steel Almirahs has fixed costs of Rs 1,00,000 and variable cost of Rs 400 equation relating costs to production. What is the cost of producing 100 almirahs?
(ii) A hotel banquets for groups of people at a cost of Rs 200 per person plus an overhead charge of 750. Find the cost C(x) they would charge for catering for x people.
(iii) A company finds its cost function to be C(x) = 100 + 50 x and its demand function to be p(x) = 102 – x. Find (a) the revenue function, (b) the profit function.
(iv) A company produced commodity with Rs 10,000 fixed costs. The variable costs are estimated to 25% of the total revenue received on selling the product at a rate of Rs 6 per unit. Find the total revenue, total cost and profit functions.
(v) The demand function for a certain commodity is given by p = 1000 – 15x − x², 0 < x < 25. What is the price per unit and the total revenue from the sale of 2 units?
Answer:
(i) Let x represent the number of steel almirahs made.
The given Total Fixed Cost (TFC) is Rs 1,00,000, and the Total Variable Cost (TVC) is Rs 400 multiplied by x.
So, the Total Cost (TC) is TFC plus TVC, which is \( 1,00,000 + 400x \). This gives us the cost function \( C(x) \).
To find the cost of producing 100 almirahs, we put \( x = 100 \) into the cost function:
\( C(100) = 100000 + 400 \times 100 \)
\( C(100) = 100000 + 40000 \)
\( C(100) = Rs 1,40,000 \). The total cost includes both fixed expenses and costs that change with production volume.
(ii) Let x be the number of people in the group.
The Total Variable Cost (TVC) is Rs 200 per person, so for x people, it is \( 200 \times x = Rs 200x \).
The Total Fixed Cost (TFC) is the overhead charge, which is Rs 750.
Therefore, the total cost function \( C(x) \) is the sum of TVC and TFC: \( C(x) = 200x + 750 \). This formula helps determine the bill for any group size.
(iii) We are given the cost function \( C(x) = 100 + 50x \) and the demand function \( p(x) = 102 - x \).
(a) The Revenue function \( R(x) \) is found by multiplying the price per unit \( p(x) \) by the number of units \( x \). So, \( R(x) = p(x) \times x = (102 - x)x = 102x - x^2 \). Revenue represents the total money earned from sales.
(b) The Profit function \( P(x) \) is calculated by subtracting the Cost function \( C(x) \) from the Revenue function \( R(x) \).
\( P(x) = R(x) - C(x) \)
\( \implies P(x) = (102x - x^2) - (100 + 50x) \)
\( \implies P(x) = 102x - x^2 - 100 - 50x \)
\( \implies P(x) = 52x - x^2 - 100 \). Profit shows how much money is left after all costs are paid.
(iv) The Total Fixed Cost (TFC) is given as Rs 10,000.
Let x be the number of units produced.
The selling price per unit is Rs 6. So, the Revenue function \( R(x) \) for x units is \( R(x) = 6x \). This is the money collected from selling the goods.
The variable costs are 25% of the total revenue. So, the Total Variable Cost (TVC) is \( 0.25 \times R(x) = 0.25 \times 6x = 1.5x \) or \( \frac{3x}{2} \). Variable costs increase as more items are produced.
The Total Cost function \( C(x) \) is TFC plus TVC: \( C(x) = 10000 + \frac{3x}{2} \).
Finally, the Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = 6x - (10000 + \frac{3x}{2}) \)
\( \implies P(x) = 6x - 10000 - \frac{3x}{2} \)
\( \implies P(x) = \frac{12x - 3x}{2} - 10000 \)
\( \implies P(x) = \frac{9x}{2} - 10000 \). A positive profit means the company is making money.
(v) We are given the demand function \( p(x) = 1000 - 15x - x^2 \), which tells us the price per unit based on the number of units x, where \( 0 < x < 25 \).
The Revenue function \( R(x) \) is found by multiplying the price per unit \( p(x) \) by the number of units \( x \): \( R(x) = p(x) \times x = (1000 - 15x - x^2)x = 1000x - 15x^2 - x^3 \).
To find the price and total revenue from the sale of 2 units, we first find \( p(2) \) and \( R(2) \).
Price per unit when \( x = 2 \):
\( p(2) = 1000 - 15(2) - (2)^2 \)
\( p(2) = 1000 - 30 - 4 \)
\( p(2) = 966 \). So, the price per unit is Rs 966.
Total Revenue for \( x = 2 \) units:
\( R(2) = 1000(2) - 15(2)^2 - (2)^3 \)
\( R(2) = 2000 - 15(4) - 8 \)
\( R(2) = 2000 - 60 - 8 \)
\( R(2) = Rs 1932 \). This is the total money earned from selling two units.
In simple words: For part (i), add the fixed cost to the variable cost for 100 almirahs. For part (ii), multiply the cost per person by x and add the fixed overhead. For part (iii), revenue is price times quantity, and profit is revenue minus cost. For part (iv), calculate variable cost as 25% of revenue before finding total cost and profit. For part (v), substitute x=2 into the demand function for price and into the revenue function for total revenue.

🎯 Exam Tip: Always clearly define what \( x \), \( p(x) \), \( C(x) \), \( R(x) \), and \( P(x) \) represent in your solutions. Pay close attention to whether costs are fixed, variable per unit, or a percentage of revenue.

Question 2.Suppose the cost to produce some commodity is a linear function of output. Find cost as a function of output, if costs are Rs 4000 for 250 units and Rs 5000 for 350 units.
Answer:Let the cost function be a linear function of x, represented as \( C(x) = ax + b \), where 'a' is the variable cost per unit and 'b' is the fixed cost. We are given that the cost for 250 units is Rs 4000. So, substituting into the equation: \( 4000 = 250a + b \) ...(1)
We are also given that the cost for 350 units is Rs 5000. So: \( 5000 = 350a + b \) ...(2)
Now, we subtract equation (1) from equation (2) to find 'a': \( (5000 - 4000) = (350a - 250a) + (b - b) \)
\( 1000 = 100a \)
\( \implies a = 10 \)
Next, substitute \( a = 10 \) back into equation (1) to find 'b': \( 4000 = 250(10) + b \)
\( 4000 = 2500 + b \)
\( \implies b = 4000 - 2500 \)
\( \implies b = 1500 \)
Therefore, the cost function is \( C(x) = 10x + 1500 \). This linear model effectively captures the cost behavior over a range of production volumes.In simple words: Since the cost changes linearly, we use the formula \( C(x) = ax + b \). We use the two given cost points to find the values of 'a' and 'b' by solving two simple equations.

🎯 Exam Tip: When given two data points for a linear cost function, set up two simultaneous equations to solve for the slope (variable cost per unit) and the y-intercept (fixed cost).

Question 3.A manufacturer can sell x items of commodity at price of (330 – x) each. Find the revenue function. If the cost of producing x items is (x² + 10x + 12), determine the profit function.
Answer:The selling price per item is given as \( (330 - x) \).
The Revenue function \( R(x) \) is the selling price per item multiplied by the number of items sold (x): \( R(x) = (330 - x)x \)
\( \implies R(x) = 330x - x^2 \). This represents the total income from selling x items.
The Cost function \( C(x) \) for producing x items is given as \( C(x) = x^2 + 10x + 12 \).
The Profit function \( P(x) \) is calculated by subtracting the Cost function from the Revenue function: \( P(x) = R(x) - C(x) \)
\( \implies P(x) = (330x - x^2) - (x^2 + 10x + 12) \)
\( \implies P(x) = 330x - x^2 - x^2 - 10x - 12 \)
\( \implies P(x) = 320x - 2x^2 - 12 \). A well-defined profit function is key to business analysis.In simple words: First, find the total money made from sales (revenue) by multiplying the price per item by the number of items. Then, subtract the total cost of making those items from the revenue to get the profit.

🎯 Exam Tip: Always be careful with algebraic signs, especially when subtracting a cost function that has multiple terms. Ensure all terms are correctly distributed.

Question 4.
(i) The fixed cost and the variable cost of x units of a manufactured product of a company are Rs 4,00,000 and Rs 80x respectively. If each unit is sold for Rs 280, what is the breakeven point?
(ii) A textbook publisher finds that the production costs to each book are 25 and that the fixed costs are Rs 15,000. If each book can be sold for Rs 45, determine (a) the cost function, (b) the revenue function, and (c) the break-even point.
(iii) The cost of producing x items per day is given in rupees as C(x) = 2000 + 100√x. If each item can be sold for Rs 10, determine the break-even point.
(iv) A television manufacturer finds that the total cost for producing and marketing x television sets is C(x)=250 x² + 325x + 10,000. Each product is sold for Rs 6,500. Determine the break-even points.
(v) A company has fixed costs of Rs 26,000. The cost one item is Rs 30. If this item sells for Rs 43, find the break-even point.
(vi) The fixed cost of new product is Rs 18,000 and the variable cost per unit is Rs 550. If the demand function is p(x) = 4000 – 150x, find the break-even values.
Answer:
(i) Given Total Fixed Cost (TFC) = Rs 4,00,000.
The Total Variable Cost (TVC) for x units is given as Rs 80x.
So, the Total Cost function \( C(x) \) is TFC + TVC: \( C(x) = 400000 + 80x \).
The selling price per unit is Rs 280. So, the Revenue function \( R(x) \) for x units is \( R(x) = 280x \).
The Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = 280x - (400000 + 80x) \)
\( \implies P(x) = 280x - 400000 - 80x \)
\( \implies P(x) = 200x - 400000 \).
For the break-even point, the profit must be zero, so \( P(x) = 0 \).
\( \implies 200x - 400000 = 0 \)
\( \implies 200x = 400000 \)
\( \implies x = \frac{400000}{200} \)
\( \implies x = 2000 \). The company needs to sell 2000 units to cover all its costs.
(ii) Given Total Fixed Cost (TFC) = Rs 15,000.
The production cost per book (variable cost per unit) is Rs 25.
(a) The Cost function \( C(x) \) for x books is TFC + (variable cost per unit \( \times \) x):
\( C(x) = 15000 + 25x \). This function shows how total costs change with the number of books.
(b) Each book is sold for Rs 45. The Revenue function \( R(x) \) for x books is selling price per unit \( \times \) x:
\( R(x) = 45x \). This is the total income from selling books.
(c) For the break-even point, Profit \( P(x) \) must be zero, which means Revenue equals Cost \( R(x) = C(x) \).
\( 45x = 15000 + 25x \)
\( \implies 45x - 25x = 15000 \)
\( \implies 20x = 15000 \)
\( \implies x = \frac{15000}{20} \)
\( \implies x = 750 \). The publisher needs to sell 750 books to break even.
(iii) Given Cost function \( C(x) = 2000 + 100\sqrt{x} \).
The selling price of each item is Rs 10. So, the Revenue function \( R(x) \) for x items is \( R(x) = 10x \).
For the break-even point, Profit \( P(x) = 0 \), meaning Revenue equals Cost: \( R(x) = C(x) \).
\( 10x = 2000 + 100\sqrt{x} \)
To solve this, we can rewrite the equation:
\( 10x - 2000 = 100\sqrt{x} \)
Divide by 10:
\( x - 200 = 10\sqrt{x} \)
Rearrange the terms:
\( x - 10\sqrt{x} - 200 = 0 \)
Let \( y = \sqrt{x} \), so \( x = y^2 \):
\( y^2 - 10y - 200 = 0 \)
This is a quadratic equation. We can solve it by factoring or using the quadratic formula.
By factoring, we need two numbers that multiply to -200 and add to -10 (which are -20 and 10):
\( (y - 20)(y + 10) = 0 \)
So, \( y = 20 \) or \( y = -10 \).
Since \( y = \sqrt{x} \), \( y \) must be positive. Thus, \( y = 20 \).
\( \sqrt{x} = 20 \)
\( \implies x = 20^2 \)
\( \implies x = 400 \). To break even, 400 items must be produced and sold.
(iv) Given Total Cost function \( C(x) = 250x^2 + 325x + 10000 \).
The selling price of each television set is Rs 6,500. So, the Revenue function \( R(x) \) for x sets is \( R(x) = 6500x \).
The Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = 6500x - (250x^2 + 325x + 10000) \)
\( \implies P(x) = 6500x - 250x^2 - 325x - 10000 \)
\( \implies P(x) = -250x^2 + 6175x - 10000 \).
For the break-even point, Profit \( P(x) = 0 \).
\( -250x^2 + 6175x - 10000 = 0 \)
Divide the equation by -25 (to simplify and make the leading coefficient positive):
\( 10x^2 - 247x + 400 = 0 \).
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 10, b = -247, c = 400 \).
\( x = \frac{-(-247) \pm \sqrt{(-247)^2 - 4(10)(400)}}{2(10)} \)
\( x = \frac{247 \pm \sqrt{61009 - 16000}}{20} \)
\( x = \frac{247 \pm \sqrt{45009}}{20} \)
\( x = \frac{247 \pm 212.1537}{20} \).
\( x_1 = \frac{247 + 212.1537}{20} = \frac{459.1537}{20} \approx 22.96 \)
\( x_2 = \frac{247 - 212.1537}{20} = \frac{34.8463}{20} \approx 1.74 \).
The break-even points are approximately 2 and 23 units. Since physical units must be whole numbers, the break-even points are at approximately 2 units and 23 units. This implies the company breaks even at two different production levels, often due to the nature of the quadratic cost function.
(v) Given Total Fixed Cost (TFC) = Rs 26,000.
The cost to produce one item (variable cost per unit) is Rs 30.
So, the Total Variable Cost (TVC) for x items is \( 30x \).
The Total Cost function \( C(x) \) is TFC + TVC: \( C(x) = 26000 + 30x \).
Each item sells for Rs 43. So, the Revenue function \( R(x) \) for x items is \( R(x) = 43x \).
The Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = 43x - (26000 + 30x) \)
\( \implies P(x) = 43x - 26000 - 30x \)
\( \implies P(x) = 13x - 26000 \).
For the break-even point, Profit \( P(x) = 0 \).
\( 13x - 26000 = 0 \)
\( \implies 13x = 26000 \)
\( \implies x = \frac{26000}{13} \)
\( \implies x = 2000 \). The company needs to sell 2000 units to cover its costs.
(vi) Given Total Fixed Cost (TFC) = Rs 18,000.
The variable cost per unit is Rs 550. So, the Total Variable Cost (TVC) for x units is \( 550x \).
The Total Cost function \( C(x) \) is TFC + TVC: \( C(x) = 18000 + 550x \).
The demand function is \( p(x) = 4000 - 150x \).
The Revenue function \( R(x) \) is price per unit \( p(x) \) multiplied by x:
\( R(x) = (4000 - 150x)x = 4000x - 150x^2 \).
The Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = (4000x - 150x^2) - (18000 + 550x) \)
\( \implies P(x) = 4000x - 150x^2 - 18000 - 550x \)
\( \implies P(x) = -150x^2 + 3450x - 18000 \).
For the break-even point, Profit \( P(x) = 0 \).
\( -150x^2 + 3450x - 18000 = 0 \)
Divide the equation by -50 to simplify:
\( 3x^2 - 69x + 360 = 0 \).
Divide by 3:
\( x^2 - 23x + 120 = 0 \).
This is a quadratic equation. We can solve it by factoring or using the quadratic formula.
By factoring, we look for two numbers that multiply to 120 and add to -23 (which are -8 and -15):
\( (x - 8)(x - 15) = 0 \)
So, \( x = 8 \) or \( x = 15 \).
The break-even points are 8 units and 15 units. These are the two production levels where the company's total costs equal its total revenue.In simple words: For break-even points, set the profit function to zero (meaning Revenue = Cost) and solve for x. Remember that profit is total revenue minus total cost.

🎯 Exam Tip: When dealing with quadratic profit functions, there can be two break-even points, indicating a range of profitable operations between them. Always check your solutions for real-world applicability (e.g., non-negative units).

Question 5.A profit making company wants to launch a new product. It observes that the fixed cost of the new product is Rs 35,000 and the variable cost per unit is Rs 500. The revenue received on the sale of x units is given by 5000 x – 100x. Find (i) profit function (ii) break-even points.
Answer:
(i) Given Total Fixed Cost (TFC) = Rs 35,000.
The variable cost per unit is Rs 500. So, the Total Variable Cost (TVC) for x units is \( 500x \).
The required Cost function \( C(x) \) is TFC + TVC: \( C(x) = 35000 + 500x \).
The Revenue function \( R(x) \) is given as \( R(x) = 5000x - 100x^2 \). This expression accounts for changes in price based on demand.
The Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = (5000x - 100x^2) - (35000 + 500x) \)
\( \implies P(x) = 5000x - 100x^2 - 35000 - 500x \)
\( \implies P(x) = -100x^2 + 4500x - 35000 \).
(ii) For the break-even point, Profit \( P(x) = 0 \).
\( -100x^2 + 4500x - 35000 = 0 \)
Divide the equation by -100 to simplify:
\( x^2 - 45x + 350 = 0 \).
This is a quadratic equation. We can solve it by factoring or using the quadratic formula.
By factoring, we look for two numbers that multiply to 350 and add to -45 (which are -10 and -35):
\( (x - 10)(x - 35) = 0 \)
So, \( x = 10 \) or \( x = 35 \).
The break-even points are 10 units and 35 units. This means the company makes zero profit at these two production levels.
(iii) For loss, Profit \( P(x) < 0 \).
This means \( -100x^2 + 4500x - 35000 < 0 \).
Dividing by -100 and reversing the inequality sign:
\( x^2 - 45x + 350 > 0 \).
Since \( (x - 10)(x - 35) = x^2 - 45x + 350 \), the inequality becomes:
\( (x - 10)(x - 35) > 0 \).
This inequality holds true if both factors are positive or both are negative.
Case 1: Both factors are positive.
\( x - 10 > 0 \implies x > 10 \)
\( x - 35 > 0 \implies x > 35 \)
For both to be true, \( x > 35 \).
Case 2: Both factors are negative.
\( x - 10 < 0 \implies x < 10 \)
\( x - 35 < 0 \implies x < 35 \)
For both to be true, \( x < 10 \).
So, the company will incur a loss if \( x > 35 \) or \( x < 10 \).In simple words: First, create the profit formula by subtracting total cost from total revenue. Then, to find the break-even points, set this profit formula to zero and solve for x. To find when there's a loss, set the profit formula to be less than zero and solve for x.

🎯 Exam Tip: When solving quadratic inequalities for profit/loss, remember to test intervals or consider the parabola's shape to determine where the function is positive (profit) or negative (loss).

Question 6.A company has fixed cost of Rs 10,000 and cost of producing one unit of its product is Rs 50. If each unit sells for 75, find the break-even value. Also, find the values of x for which the company results in profit.
Answer:Given total fixed cost (TFC) = Rs 10,000.
The cost of producing one unit (variable cost per unit) is Rs 50.
So, the Total Variable Cost (TVC) for producing x units is \( 50x \).
The Total Cost function \( C(x) \) is TFC + TVC: \( C(x) = 10000 + 50x \).
Each unit sells for Rs 75. So, the Revenue function \( R(x) \) for x units is \( R(x) = 75x \).
The Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = 75x - (10000 + 50x) \)
\( \implies P(x) = 75x - 10000 - 50x \)
\( \implies P(x) = 25x - 10000 \).
For the break-even point, Profit \( P(x) = 0 \).
\( 25x - 10000 = 0 \)
\( \implies 25x = 10000 \)
\( \implies x = \frac{10000}{25} \)
\( \implies x = 400 \). The break-even value is 400 units.
For the company to make a profit, the Profit function \( P(x) \) must be greater than zero: \( P(x) > 0 \).
\( 25x - 10000 > 0 \)
\( \implies 25x > 10000 \)
\( \implies x > \frac{10000}{25} \)
\( \implies x > 400 \). This shows that producing more than 400 units will lead to a profit.
Thus, the company will make a profit if it produces and sells more than 400 units of the product.In simple words: To find when the company breaks even, calculate the profit function (revenue minus cost) and set it to zero. To find when the company makes a profit, set the profit function to be greater than zero.

🎯 Exam Tip: Be sure to distinguish between break-even (profit = 0) and making a profit (profit > 0). These questions often require solving both an equation and an inequality.

Question 7.
(i) A company sells its products at Rs 10 per unit. Fixed costs for the company are Rs 35,000 and variable costs are estimated to run 30% of total revenue. Determiae the (a) total revenue function (b) total cost function (c) quantity the company must sell to cover the fixed cost.
(ii) The price of selling one unit of a product when x units are demanded given by the equation p =4000 – 2x. The fixed costs of the product are Rs 20,000 and Rs 148 per unit are paid for the product to place in a store. Find the level of sales at which the company expects to cover its costs.
Answer:
(i) Let x be the number of products sold.
(a) The selling price per unit is Rs 10. So, the Total Revenue function \( R(x) \) for x units is \( R(x) = 10x \). This represents the total income from sales.
(b) Given Total Fixed Cost (TFC) = Rs 35,000.
Variable costs are 30% of total revenue. So, Total Variable Cost (TVC) = \( 30\% \) of \( R(x) \).
\( \text{TVC} = \frac{30}{100} \times 10x = 3x \). Variable costs change with the number of units sold.
The Total Cost function \( C(x) \) is TFC + TVC: \( C(x) = 35000 + 3x \).
(c) To cover fixed costs, the company needs enough revenue to equal its total variable costs plus fixed costs, but "cover fixed cost" likely implies a partial break-even or a very simplified calculation for a specific scenario. However, the question seems to be asking for the general quantity to *cover costs* which is the break-even point.
The question phrasing "quantity the company must sell to cover the fixed cost" is ambiguous. If it literally means to cover *only* fixed costs from revenue, while variable costs are considered separately or non-existent in this context, it would be \( 10x = 35000 \). But with variable costs existing, the standard interpretation is break-even.
Let's assume the question means to find x such that the revenue covers *all* costs (the break-even point). Then \( R(x) = C(x) \).
\( 10x = 35000 + 3x \)
\( \implies 10x - 3x = 35000 \)
\( \implies 7x = 35000 \)
\( \implies x = \frac{35000}{7} \)
\( \implies x = 5000 \). So, the company must sell 5000 units to cover all its costs. (ii) Given selling price of each unit of product \( p(x) = 4000 - 2x \).
The Revenue function \( R(x) \) is price per unit \( p(x) \) multiplied by x: \( R(x) = (4000 - 2x)x = 4000x - 2x^2 \).
Given Total Fixed Costs (TFC) = Rs 20,000.
Variable costs per unit (for placing in store) = Rs 148.
So, Total Variable Cost (TVC) for x units = \( 148x \).
The Total Cost function \( C(x) \) is TFC + TVC: \( C(x) = 20000 + 148x \).
To find the level of sales at which the company expects to cover its costs (break-even point), we set the Profit function \( P(x) = 0 \), which means \( R(x) = C(x) \).
\( 4000x - 2x^2 = 20000 + 148x \)
Rearrange the terms to form a quadratic equation:
\( 0 = 2x^2 + 148x - 4000x + 20000 \)
\( 0 = 2x^2 - 3852x + 20000 \)
Divide by 2:
\( x^2 - 1926x + 10000 = 0 \).
This quadratic equation needs to be solved for x using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
Here, \( a=1, b=-1926, c=10000 \).
\( x = \frac{-(-1926) \pm \sqrt{(-1926)^2 - 4(1)(10000)}}{2(1)} \)
\( x = \frac{1926 \pm \sqrt{3709076 - 40000}}{2} \)
\( x = \frac{1926 \pm \sqrt{3669076}}{2} \)
\( x = \frac{1926 \pm 1915.483}{2} \).
\( x_1 = \frac{1926 + 1915.483}{2} = \frac{3841.483}{2} \approx 1920.74 \)
\( x_2 = \frac{1926 - 1915.483}{2} = \frac{10.517}{2} \approx 5.26 \).
Since the company expects to cover its costs, we are looking for valid production levels. The break-even points are approximately 5 units and 1921 units.In simple words: For part (i), calculate revenue as 10x, variable cost as 30% of revenue, and total cost as fixed cost plus variable cost. Set revenue equal to total cost to find the break-even quantity. For part (ii), use the demand function to get revenue, and sum fixed and variable costs for total cost. Set revenue equal to total cost and solve the resulting quadratic equation to find the break-even sales levels.

🎯 Exam Tip: Pay close attention to how variable costs are defined (per unit or as a percentage of revenue). When solving quadratic equations for break-even points, both valid positive solutions should be reported.

Question 8.A computer software manufacturer wants to start the production of floppy disks. He observes that he will have to spend Rs 2 lakh for the technical know-how. The cost of setting up the machine is Rs 88,000 and the cost of producing each unit is Rs 30. He can sell each floppy at Rs 45. Determine (i) the total cost function for producing x floppies, and (ii) the break-even point.
Answer:Given total money spent for technical know-how = Rs 2,00,000.
Cost of setting up the machine = Rs 88,000.
(i) The Total Fixed Cost (TFC) is the sum of technical know-how and machine setup cost:
\( \text{TFC} = 200000 + 88000 = Rs 2,88,000 \).
The cost of producing each unit (variable cost per unit) is Rs 30. So, the Total Variable Cost (TVC) for x floppies is \( 30x \).
The Total Cost function \( C(x) \) for producing x floppies is TFC + TVC:
\( C(x) = 288000 + 30x \). This function helps in planning production expenses.
(ii) The selling price of one floppy is Rs 45.
So, the Revenue function \( R(x) \) for x floppies is \( R(x) = 45x \).
The Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = 45x - (288000 + 30x) \)
\( \implies P(x) = 45x - 288000 - 30x \)
\( \implies P(x) = 15x - 288000 \).
For the break-even point, Profit \( P(x) = 0 \).
\( 15x - 288000 = 0 \)
\( \implies 15x = 288000 \)
\( \implies x = \frac{288000}{15} \)
\( \implies x = 19200 \).
Hence, the company must sell at least 19200 units to cover its costs.In simple words: First, add up all the fixed costs (know-how and machine setup) to get the total fixed cost. Then, add the variable cost (30 times x) to the fixed cost to get the total cost function. Finally, set the total revenue (45 times x) equal to the total cost to find how many units need to be sold to break even.

🎯 Exam Tip: Ensure all initial setup costs, even those like 'technical know-how', are correctly categorized as fixed costs when formulating the total cost function.

Question 9.A company sells its product at the rate of Rs 6 per unit. The variable costs are estimated to run 25% of the total revenue received. If the fixed costs for the product are 45,00, find (i) The total revenue function, (ii) The total cost function, (iii) The profit function, (iv) The break-even point, (v) The number of units the company must sell to cover its fixed cost.
Answer:Given Total Fixed Cost (TFC) = Rs 4500 (assuming "45,00" meant 4500 based on the solution context where it's used as 4500).
The selling price of one unit = Rs 6.
(i) The Revenue function \( R(x) \) for x units is the selling price per unit multiplied by x:
\( R(x) = 6x \). This shows the total money earned from sales.
(ii) The variable costs are 25% of the total revenue received.
So, Total Variable Cost (TVC) = \( 25\% \) of \( R(x) \)
\( \text{TVC} = \frac{1}{4} \times 6x = \frac{3x}{2} \). Variable costs increase directly with the number of units sold.
The Total Cost function \( C(x) \) is TFC + TVC:
\( C(x) = 4500 + \frac{3x}{2} \).
(iii) The Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = 6x - (4500 + \frac{3x}{2}) \)
\( \implies P(x) = 6x - 4500 - \frac{3x}{2} \)
\( \implies P(x) = \frac{12x - 3x}{2} - 4500 \)
\( \implies P(x) = \frac{9x}{2} - 4500 \). A positive profit means the company is doing well.
(iv) For the break-even point, Profit \( P(x) = 0 \).
\( \frac{9x}{2} - 4500 = 0 \)
\( \implies \frac{9x}{2} = 4500 \)
\( \implies 9x = 4500 \times 2 \)
\( \implies 9x = 9000 \)
\( \implies x = \frac{9000}{9} \)
\( \implies x = 1000 \). The company needs to sell 1000 units to break even.
(v) The number of units the company must sell to cover its fixed cost. This is the same as the break-even point, where total revenue covers total costs, including fixed costs.
If the question specifically refers to covering *only* fixed costs, it would imply setting total revenue equal to fixed costs: \( 6x = 4500 \).
\( \implies x = \frac{4500}{6} \)
\( \implies x = 750 \). This value tells us how many units need to be sold just to make up for the fixed expenses, not accounting for variable costs in this specific interpretation.In simple words: First, write down the formulas for revenue, variable cost (25% of revenue), total cost (fixed + variable), and profit (revenue - total cost). To break even, set profit to zero. To cover fixed costs, find how many units generate enough revenue to match the fixed costs.

🎯 Exam Tip: Carefully read questions asking about "covering costs." If variable costs are present, "covering costs" generally implies the break-even point. If it asks specifically to "cover fixed costs" and variable costs are present, it might be a trick question or a simplified scenario for just fixed cost recovery.

Question 10.A company sells x tins of talcum powder each day as 10 per tin. The cost of mauufacturing is 6 per tin and the distributor charges 1 per tin. Besides these daily overhead cost, a fixed cost comes to Rs 600. Determine the profit function. What is the profit if 500 tins are manufactured and sold a day. How do you interpret the situation if the company manufactures and sells 100 tins a day? What is the break-even point?
Answer:The selling price (S.P.) of one tin of talcum powder = Rs 10.
The S.P. of x tins of talcum powder (Revenue function \( R(x) \)) = \( 10x \).
The cost of manufacturing one tin = Rs 6.
The distributor charge per tin = Rs 1.
So, the total variable cost price of one tin = Rs \( (6 + 1) = Rs 7 \).
The Total Variable Cost (TVC) for x tins = \( 7x \).
The Fixed Cost (TFC) = Rs 600.
The Total Cost function \( C(x) \) is TFC + TVC: \( C(x) = 600 + 7x \).
The Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = 10x - (600 + 7x) \)
\( \implies P(x) = 10x - 600 - 7x \)
\( \implies P(x) = 3x - 600 \). This function describes how profit changes with sales.
To find the profit if 500 tins are manufactured and sold a day, substitute \( x = 500 \) into the profit function:
\( P(500) = 3 \times 500 - 600 \)
\( P(500) = 1500 - 600 \)
\( P(500) = Rs 900 \). The company makes a profit of Rs 900 if 500 tins are sold.
If the company manufactures and sells 100 tins per day, substitute \( x = 100 \) into the profit function:
\( P(100) = 3 \times 100 - 600 \)
\( P(100) = 300 - 600 \)
\( P(100) = -300 \).
This means the company has a loss of Rs 300 if 100 tins are sold. This indicates that at this production level, the revenue is not enough to cover all costs.
For the break-even point, Profit \( P(x) = 0 \).
\( 3x - 600 = 0 \)
\( \implies 3x = 600 \)
\( \implies x = \frac{600}{3} \)
\( \implies x = 200 \). The break-even point is 200 tins.In simple words: First, figure out the total variable cost per tin by adding manufacturing cost and distributor charge. Then, create the total cost function (fixed cost plus total variable cost for x tins) and the revenue function (selling price per tin times x). The profit function is revenue minus cost. Use this to calculate profit at 500 and 100 tins, and find the break-even point by setting profit to zero.

🎯 Exam Tip: When multiple costs per unit are given, add them all to get the total variable cost per unit. Negative profit means a loss, and positive profit means the business is earning money.

Question 11.From the following information, calculate break-even point in terms of sales value and in units. Fixed factory overhead cost = Rs 60,000. Fixed selling overhead cost = Rs 12,000, variable manufacturing cost = Rs 12 per unit, selling price per unit = Rs 24.
Answer:Fixed factory overhead cost = Rs 60,000.
Fixed selling overhead cost = Rs 12,000.
The Total Fixed Cost (TFC) is the sum of all fixed costs:
\( \text{TFC} = 60000 + 12000 = Rs 72,000 \). Fixed costs do not change with the number of units produced.
Variable manufacturing cost per unit = Rs 12.
So, the Total Variable Cost (TVC) for x units is \( 12x \).
The Total Cost function \( C(x) \) is TFC + TVC: \( C(x) = 72000 + 12x \).
The selling price per unit = Rs 24.
So, the Revenue function \( R(x) \) for x units is \( R(x) = 24x \).
The Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = 24x - (72000 + 12x) \)
\( \implies P(x) = 24x - 72000 - 12x \)
\( \implies P(x) = 12x - 72000 \).
For the break-even point in units, Profit \( P(x) = 0 \).
\( 12x - 72000 = 0 \)
\( \implies 12x = 72000 \)
\( \implies x = \frac{72000}{12} \)
\( \implies x = 6000 \). The break-even point is 6000 units.
To find the break-even point in terms of sales value, multiply the break-even units by the selling price per unit:
Sales value at break-even point = Break-even units \( \times \) Selling price per unit
Sales value = \( 6000 \times Rs 24 = Rs 1,44,000 \). This is the total revenue needed to cover all costs.In simple words: Add all fixed costs to get total fixed cost. Then, write the total cost function (total fixed cost plus variable cost per unit times x) and the revenue function (selling price per unit times x). To find the break-even point in units, set the profit to zero and solve for x. To find the break-even sales value, multiply the break-even units by the selling price per unit.

🎯 Exam Tip: Remember to sum all components of fixed costs to get the total fixed cost. Break-even can be expressed both in units and in total sales value; make sure to calculate both if asked.

Question 12.The daily cost of production C for x units of an assembly is given by C(x) = (12.5x + 6400)
(i) If each unit is sold, for Rs 25, determine the minimum number of units that should be produced and sold to ensure no loss.
(ii) If the selling price is reduced by 2.50 per unit, what would be the break-even point?
(iii) If it is known that 500 units can be sold daily what price per unit should be charged to guarantee no loss?
Answer:Given cost function \( C(x) = 12.5x + 6400 \). Here, Rs 6400 is the fixed cost and Rs 12.5 is the variable cost per unit.
(i) If each unit is sold for Rs 25, the Revenue function \( R(x) \) for x units is \( R(x) = 25x \).
To ensure no loss, the company must reach the break-even point where Profit \( P(x) = 0 \), meaning Revenue equals Cost: \( R(x) = C(x) \).
\( 25x = 12.5x + 6400 \)
\( \implies 25x - 12.5x = 6400 \)
\( \implies 12.5x = 6400 \)
\( \implies x = \frac{6400}{12.5} \)
\( \implies x = 512 \).
The minimum number of units that should be produced and sold to ensure no loss is 512 units. This level covers all expenses.
(ii) If the selling price is reduced by Rs 2.50 per unit, the new selling price per unit will be \( Rs (25 - 2.50) = Rs 22.50 \).
The new Revenue function \( R'(x) \) for x units is \( R'(x) = 22.50x \).
For the new break-even point, \( R'(x) = C(x) \).
\( 22.50x = 12.5x + 6400 \)
\( \implies 22.50x - 12.5x = 6400 \)
\( \implies 10x = 6400 \)
\( \implies x = \frac{6400}{10} \)
\( \implies x = 640 \).
If the selling price is reduced, the break-even point increases to 640 units.
(iii) If 500 units can be sold daily, so \( x = 500 \). To guarantee no loss, the total revenue from 500 units must be equal to or greater than the total cost of 500 units.
Let the selling price per unit be 'y'. Then the Revenue function is \( R(x) = yx \). For 500 units, \( R(500) = 500y \).
First, calculate the Total Cost for 500 units using the given cost function:
\( C(500) = 12.5(500) + 6400 \)
\( C(500) = 6250 + 6400 \)
\( C(500) = Rs 12,650 \). This is the total expense for producing 500 units.
For no loss (break-even at 500 units), Revenue must equal Cost:
\( R(500) = C(500) \)
\( 500y = 12650 \)
\( \implies y = \frac{12650}{500} \)
\( \implies y = 25.30 \).
Hence, the required price per unit should be Rs 25.30 to guarantee no loss if 500 units are sold.In simple words: For (i), set revenue (25x) equal to cost (12.5x + 6400) and solve for x. For (ii), first find the new selling price (25 - 2.50), then set the new revenue equal to the cost and solve for x. For (iii), calculate the total cost for 500 units, then set the revenue for 500 units (500y) equal to this cost to find the required selling price 'y'.

🎯 Exam Tip: Always clearly state the conditions (original price, reduced price, or target quantity) before calculating the break-even point or required price. Ensure your calculations are consistent with the scenario described.

Question 13.A company decides to set up a small production plant for manufacturing electronic clocks. The total cost for initial set up as fixed cost is Rs 9 lakh. The additional cost (i.e., variable cost) for producing each clock is Rs 300. Each clock is sold at Rs 750. During the first month 1500 clocks are produced and sold.
(i) Determine the cost function C(x) for the total cost of producing x clocks.
(ii) Determine the revenue function R(x) for the total revenue from the sale of x clocks.
(iii) Determine the profit function P(x) for the total revenue from the sale of x clocks.
(iv) What profit or loss the company incurs during the first month when all the 1500 clocks are sold.
(v) Determine the break-even point.
Answer:Given total fixed cost (TFC) = Rs 9 lakh = Rs 9,00,000.
The cost of producing each clock (variable cost per unit) = Rs 300.
(i) The cost of producing x clocks (Total Variable Cost, TVC) = \( 300x \).
The Total Cost function \( C(x) \) is TFC + TVC: \( C(x) = 900000 + 300x \). This function describes all the expenses involved.
(ii) The selling price of each clock = Rs 750.
The Revenue function \( R(x) \) for x clocks is the selling price per unit multiplied by x:
\( R(x) = 750x \). This is the total income from selling x clocks.
(iii) The Profit function \( P(x) \) is Revenue minus Cost: \( P(x) = R(x) - C(x) \).
\( \implies P(x) = 750x - (900000 + 300x) \)
\( \implies P(x) = 750x - 900000 - 300x \)
\( \implies P(x) = 450x - 900000 \). A positive profit value indicates successful operations.
(iv) During the first month, 1500 clocks are produced and sold. To find the profit or loss, substitute \( x = 1500 \) into the Profit function:
\( P(1500) = 450 \times 1500 - 900000 \)
\( P(1500) = 675000 - 900000 \)
\( P(1500) = -2,25,000 \).
Clearly, the company incurs a loss of Rs 2,25,000 during the first month when all 1500 clocks are sold. This is because the revenue from 1500 clocks is not enough to cover the high fixed costs yet.
(v) For the break-even point, Profit \( P(x) = 0 \).
\( 450x - 900000 = 0 \)
\( \implies 450x = 900000 \)
\( \implies x = \frac{900000}{450} \)
\( \implies x = 2000 \).
The break-even point is 2000 units.In simple words: For (i) and (ii), write the cost and revenue functions using given fixed costs, variable cost per unit, and selling price per unit. For (iii), subtract the cost function from the revenue function to get the profit function. For (iv), plug 1500 units into the profit function to see if it's a gain or loss. For (v), set the profit function to zero to find the break-even number of units.

🎯 Exam Tip: Always convert large numbers (like 9 lakh) into numerical format (9,00,000) for accurate calculations. A negative profit value directly indicates a loss, and the amount of loss is the absolute value of that negative profit.

Question 14.The pricing policy of a company follows the demand function p = D(x), D(x) being the price per unit when x units are demanded. After studying the market trends the company determines the price function that is given by D(x) = 2000 – 4x.
If the product is to be marketed the company will incur a fixed cost of Rs 60,000 and will have to pay 600 for each unit that is produced and placed in the store. At what sales level can the company expect to recover its costs?
Answer:Given demand function \( D(x) = 2000 - 4x \). This function determines the price per unit based on the quantity demanded.
The Revenue function \( R(x) \) is the price per unit \( D(x) \) multiplied by x:
\( R(x) = D(x) \times x = (2000 - 4x)x = 2000x - 4x^2 \).
Given Total Fixed Cost (TFC) = Rs 60,000.
The cost for each unit produced and placed in the store (variable cost per unit) = Rs 600.
So, the Total Variable Cost (TVC) for x units is \( 600x \).
The Total Cost function \( C(x) \) is TFC + TVC: \( C(x) = 60000 + 600x \).
To find the sales level at which the company expects to recover its costs (break-even point), we set Revenue equal to Cost: \( R(x) = C(x) \).
\( 2000x - 4x^2 = 60000 + 600x \)
Rearrange the terms to form a quadratic equation:
\( 0 = 4x^2 + 600x - 2000x + 60000 \)
\( 0 = 4x^2 - 1400x + 60000 \)
Divide the equation by 4 to simplify:
\( x^2 - 350x + 15000 = 0 \).
This is a quadratic equation. We can solve it by factoring or using the quadratic formula.
By factoring, we look for two numbers that multiply to 15000 and add to -350 (which are -50 and -300):
\( (x - 300)(x - 50) = 0 \)
So, \( x = 300 \) or \( x = 50 \).
The company can expect to recover its costs at sales levels of 50 units or 300 units. These two points define the range where the business is profitable.In simple words: First, find the revenue function by multiplying the demand function by x. Then, find the total cost function by adding fixed costs and variable costs. To recover costs, set the revenue equal to the total cost and solve the resulting quadratic equation for x to find the two break-even points.

🎯 Exam Tip: When the demand function is linear, the revenue function will be quadratic. Be prepared to solve quadratic equations to find break-even points, which often result in two values for quantity.