Samacheer Kalvi Class 12 Business Maths Solutions Chapter 3 Integral Calculus II Exercise 3.3

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 3 Integral Calculus II Ex 3.3

Question 1. Calculate consumer's surplus if the demand function p = 50 – 2x and x = 20
Answer:
Given the demand function \( p = 50 - 2x \) and \( x = 20 \).
When \( x = 20 \), substitute this value into the demand function to find the price:
\( p = 50 - 2(20) \)
\( p = 50 - 40 \)
\( p = 10 \)
So, the equilibrium price \( p_0 = 10 \).
The formula for Consumer's Surplus (C.S) is:
\( C.S = \int_{0}^{x_0} f(x) \, dx - (p_0 \times x_0) \)
Substitute the given values into the formula:
\[ C.S = \int_{0}^{20} (50 - 2x) \, dx - (10 \times 20) \] First, integrate the demand function:
\[ \int (50 - 2x) \, dx = 50x - 2\left(\frac{x^2}{2}\right) = 50x - x^2 \] Now, apply the limits of integration from 0 to 20:
\[ [50x - x^2]_{0}^{20} - 200 \] \[ = \{50(20) - (20)^2\} - \{50(0) - (0)^2\} - 200 \] \[ = (1000 - 400) - 0 - 200 \] \[ = 600 - 200 \] \[ = 400 \] Therefore, the consumer's surplus is 400 units. Consumer surplus quantifies the economic benefit to consumers who can buy a product for less than the maximum they are willing to pay.
In simple words: First, we find the price when 20 units are demanded. Then, we use a special math tool called integration to find the total value consumers get, and subtract the actual money they spent. The final number shows how much extra benefit buyers received.

🎯 Exam Tip: Remember to correctly identify \( p_0 \) and \( x_0 \) and substitute them into the consumer surplus formula. Pay close attention to the limits of integration.

Question 2. Calculate consumer's surplus if the demand function p = 122 – 5x – 2x² and x = 6.
Answer:
Given the demand function \( p = 122 - 5x - 2x^2 \) and \( x = 6 \).
When \( x = 6 \), we find the corresponding price:
\( p = 122 - 5(6) - 2(6)^2 \)
\( p = 122 - 30 - 2(36) \)
\( p = 122 - 30 - 72 \)
\( p = 122 - 102 \)
\( p = 20 \)
So, the equilibrium price \( p_0 = 20 \). The given quantity is \( x_0 = 6 \).
The formula for Consumer's Surplus (C.S) is:
\( C.S = \int_{0}^{x_0} f(x) \, dx - (p_0 \times x_0) \)
Substitute the values into the formula:
\[ C.S = \int_{0}^{6} (122 - 5x - 2x^2) \, dx - (20 \times 6) \] First, integrate the demand function:
\[ \int (122 - 5x - 2x^2) \, dx = 122x - 5\left(\frac{x^2}{2}\right) - 2\left(\frac{x^3}{3}\right) \] Now, apply the limits of integration from 0 to 6:
\[ \left[122x - \frac{5x^2}{2} - \frac{2x^3}{3}\right]_{0}^{6} - 120 \] \[ = \left[122(6) - \frac{5(6)^2}{2} - \frac{2(6)^3}{3}\right] - \left[122(0) - \frac{5(0)^2}{2} - \frac{2(0)^3}{3}\right] - 120 \] \[ = \left[732 - \frac{5(36)}{2} - \frac{2(216)}{3}\right] - [0] - 120 \] \[ = \left[732 - 5(18) - 2(72)\right] - 120 \] \[ = [732 - 90 - 144] - 120 \] \[ = [732 - 234] - 120 \] \[ = 498 - 120 \] \[ = 378 \] Thus, the consumer's surplus is 378 units. This indicates the total extra value consumers gain from buying the product at the market price.
In simple words: We first find the market price for 6 units. Then, we use integration to find the total value buyers get and subtract the money they actually pay. This difference shows the extra benefit customers receive.

🎯 Exam Tip: Be careful with the calculations, especially when dealing with squares and cubes in the integration. Double-check your arithmetic to avoid errors.

Question 3. The demand function p = 85 – 5x and supply function p = 3x – 3. Calculate the equilibrium price and quantity demanded. Also calculate consumer's surplus.
Answer:
Given the demand function \( p_d = 85 - 5x \) and supply function \( p_s = 3x - 3 \).
At equilibrium, the demand price equals the supply price, so \( p_d = p_s \).
\( 85 - 5x = 3x - 3 \)
To solve for \( x \), group the constants and variables:
\( 85 + 3 = 3x + 5x \)
\( 120 = 8x \)
Divide by 8 to find \( x \):
\( x = \frac{120}{8} \)
\( x = 15 \)
So, the equilibrium quantity \( x_0 = 15 \).
Now, substitute \( x = 15 \) into either the demand or supply function to find the equilibrium price \( p_0 \). Using the demand function:
\( p_0 = 85 - 5(15) \)
\( p_0 = 85 - 75 \)
\( p_0 = 10 \)
Thus, the equilibrium quantity is 15 units and the equilibrium price is Rs 10.
Next, calculate the Consumer's Surplus (C.S). The formula is:
\( C.S = \int_{0}^{x_0} f(x) \, dx - (p_0 \times x_0) \)
Substitute \( x_0 = 15 \), \( p_0 = 10 \), and \( f(x) = 85 - 5x \):
\[ C.S = \int_{0}^{15} (85 - 5x) \, dx - (15 \times 10) \] Integrate the demand function:
\[ \int (85 - 5x) \, dx = 85x - 5\left(\frac{x^2}{2}\right) \] Apply the limits from 0 to 15:
\[ \left[85x - \frac{5x^2}{2}\right]_{0}^{15} - 150 \] \[ = \left[85(15) - \frac{5(15)^2}{2}\right] - \left[85(0) - \frac{5(0)^2}{2}\right] - 150 \] \[ = \left[1275 - \frac{5(225)}{2}\right] - 0 - 150 \] \[ = 1275 - \frac{1125}{2} - 150 \] \[ = 1275 - 562.50 - 150 \] \[ = 1275 - 712.50 \] \[ = 562.50 \] Therefore, the consumer's surplus is 562.50 units. Finding the equilibrium point is a crucial first step in market analysis.
In simple words: First, we find the point where buyers and sellers agree on a price and quantity. This is called equilibrium. Then, we use integration to calculate the extra value buyers get compared to what they actually pay.

🎯 Exam Tip: Always set demand equal to supply to find the equilibrium quantity and price first. Remember that consumer surplus is calculated using the demand function and the equilibrium values.

Question 4. The demand function for a commodity is p = e¯x. Find the consumer's surplus when p = 0.5
Answer:
Given the demand function \( p = e^{-x} \) and the market price \( p = 0.5 \).
To find the equilibrium quantity \( x_0 \), set \( p = 0.5 \):
\( 0.5 = e^{-x} \)
We can write 0.5 as \( \frac{1}{2} \).
\( \frac{1}{2} = e^{-x} \)
This means \( \frac{1}{2} = \frac{1}{e^x} \).
Taking the reciprocal of both sides gives:
\( e^x = 2 \)
Now, take the natural logarithm (ln) of both sides to solve for \( x \):
\( \ln(e^x) = \ln(2) \)
\( x = \ln(2) \)
So, the equilibrium quantity \( x_0 = \ln(2) \). The equilibrium price \( p_0 = 0.5 \).
The formula for Consumer's Surplus (C.S) is:
\( C.S = \int_{0}^{x_0} f(x) \, dx - (p_0 \times x_0) \)
Substitute the values:
\[ C.S = \int_{0}^{\ln 2} e^{-x} \, dx - (0.5 \times \ln 2) \] Integrate \( e^{-x} \):
\[ \int e^{-x} \, dx = -e^{-x} \] Apply the limits from 0 to \( \ln 2 \):
\[ [-e^{-x}]_{0}^{\ln 2} - 0.5 \ln 2 \] \[ = (-e^{-\ln 2}) - (-e^{-0}) - 0.5 \ln 2 \] \[ = (-e^{\ln (2^{-1})}) - (-e^0) - 0.5 \ln 2 \] \[ = (-2^{-1}) - (-1) - 0.5 \ln 2 \] \[ = -\frac{1}{2} + 1 - 0.5 \ln 2 \] \[ = \frac{1}{2} - 0.5 \ln 2 \] We can also write 0.5 as \( \frac{1}{2} \).
\[ C.S = \frac{1}{2} - \frac{1}{2} \ln 2 \] Factor out \( \frac{1}{2} \):
\[ C.S = \frac{1}{2} (1 - \ln 2) \] Therefore, the consumer's surplus is \( \frac{1}{2} (1 - \ln 2) \) units. Exponential demand functions are common in economic models.
In simple words: Given how the demand changes with price (using an 'e' function), and the actual price, we first find how many units are bought. Then, we use integration with logarithms to calculate the extra value customers gain by buying at that price.

🎯 Exam Tip: When dealing with exponential functions, remember to use natural logarithms to solve for \( x \). Be careful with the signs and the evaluation of \( e^0 \) and \( e^{-\ln 2} \).

Question 5. Calculate the producer's surplus at x = 5 for the supply function p = 7 + x
Answer:
Given the supply function \( p = 7 + x \) and the market quantity \( x = 5 \).
When \( x = 5 \), we find the corresponding price:
\( p = 7 + 5 \)
\( p = 12 \)
So, the equilibrium quantity \( x_0 = 5 \) and equilibrium price \( p_0 = 12 \).
The formula for Producer's Surplus (P.S) is:
\( P.S = (x_0 \times p_0) - \int_{0}^{x_0} g(x) \, dx \)
Substitute the values into the formula:
\[ P.S = (5 \times 12) - \int_{0}^{5} (7 + x) \, dx \] \[ P.S = 60 - \left[7x + \frac{x^2}{2}\right]_{0}^{5} \] First, evaluate the integral at the upper limit (5):
\[ \left[7(5) + \frac{(5)^2}{2}\right] = 35 + \frac{25}{2} \] Then, evaluate at the lower limit (0):
\[ \left[7(0) + \frac{(0)^2}{2}\right] = 0 \] Now, substitute these back into the P.S formula:
\[ P.S = 60 - \left[\left(35 + \frac{25}{2}\right) - 0\right] \] \[ P.S = 60 - \left(35 + 12.5\right) \] \[ P.S = 60 - 47.5 \] \[ P.S = 12.5 \] This can be written as a fraction:
\[ P.S = \frac{25}{2} \] Therefore, the producer's surplus is \( \frac{25}{2} \) units. Producer surplus helps economists understand the profitability of a market for sellers.
In simple words: We find the market price for 5 units supplied. Then, we calculate the total money sellers receive and subtract their minimum acceptable earnings using integration. The remaining amount is the extra profit sellers make.

🎯 Exam Tip: Remember that producer's surplus calculates the benefit to sellers, so the integration part is subtracted from the total revenue \( x_0 p_0 \). Be careful with the signs and the order of operations.

Question 6. If the supply function for a product is p = 3x + 5x². Find the producer's surplus when x = 4
Answer:
Given the supply function \( p = 3x + 5x^2 \) and the market quantity \( x = 4 \).
When \( x = 4 \), we find the corresponding price:
\( p = 3(4) + 5(4)^2 \)
\( p = 12 + 5(16) \)
\( p = 12 + 80 \)
\( p = 92 \)
So, the equilibrium quantity \( x_0 = 4 \) and equilibrium price \( p_0 = 92 \).
The formula for Producer's Surplus (P.S) is:
\( P.S = (x_0 \times p_0) - \int_{0}^{x_0} g(x) \, dx \)
Substitute the values into the formula:
\[ P.S = (4 \times 92) - \int_{0}^{4} (3x + 5x^2) \, dx \] \[ P.S = 368 - \left[\frac{3x^2}{2} + \frac{5x^3}{3}\right]_{0}^{4} \] First, evaluate the integral at the upper limit (4):
\[ \left[\frac{3(4)^2}{2} + \frac{5(4)^3}{3}\right] = \left[\frac{3(16)}{2} + \frac{5(64)}{3}\right] \] \[ = \left[\frac{48}{2} + \frac{320}{3}\right] = \left[24 + \frac{320}{3}\right] \] Then, evaluate at the lower limit (0):
\[ \left[\frac{3(0)^2}{2} + \frac{5(0)^3}{3}\right] = 0 \] Now, substitute these back into the P.S formula:
\[ P.S = 368 - \left[\left(24 + \frac{320}{3}\right) - 0\right] \] \[ P.S = 368 - \left(24 + \frac{320}{3}\right) \] To combine \( 24 + \frac{320}{3} \), find a common denominator:
\[ 24 = \frac{24 \times 3}{3} = \frac{72}{3} \] \[ P.S = 368 - \left(\frac{72}{3} + \frac{320}{3}\right) \] \[ P.S = 368 - \frac{392}{3} \] To subtract, find a common denominator for 368:
\[ 368 = \frac{368 \times 3}{3} = \frac{1104}{3} \] \[ P.S = \frac{1104}{3} - \frac{392}{3} \] \[ P.S = \frac{712}{3} \] \[ P.S \approx 237.33 \] Therefore, the producer's surplus is approximately 237.3 units. This value reflects the producers' gain from market transactions.
In simple words: We find the market price for 4 units supplied. Then, we take the total money sellers get and subtract the minimum amount they needed using integration. The result is the extra profit for the sellers.

🎯 Exam Tip: Pay attention to fractions and ensure common denominators are used correctly during subtraction. Review your integration steps carefully, especially with higher powers of x.

Question 7. The demand function for a commodity is p = \( \frac { 36 }{x+4} \) Find the producer's surplus when the prevailing market price is Rs 6.
Answer:
Given the demand function \( p = \frac{36}{x+4} \) and the market price \( p = 6 \).
To find the equilibrium quantity \( x_0 \), set \( p = 6 \):
\( 6 = \frac{36}{x+4} \)
Multiply both sides by \( (x+4) \):
\( 6(x+4) = 36 \)
Divide by 6:
\( x+4 = \frac{36}{6} \)
\( x+4 = 6 \)
Subtract 4 from both sides:
\( x = 6 - 4 \)
\( x = 2 \)
So, the equilibrium quantity \( x_0 = 2 \) and equilibrium price \( p_0 = 6 \).
The formula for Consumer's Surplus (C.S) is:
\( C.S = \int_{0}^{x_0} f(x) \, dx - (p_0 \times x_0) \)
Wait, the question asks for Producer's Surplus. It seems there is a mix-up in the problem description, as \( p = \frac{36}{x+4} \) is typically a demand function. If we assume it is a demand function and the question still asks for producer surplus, we need the supply function as well, which is not provided. Given the format, it's most likely asking for Consumer's Surplus using the given demand function and price. Let's proceed with Consumer's Surplus, assuming it was a typo in the question for "producer's surplus".
\[ C.S = \int_{0}^{2} \frac{36}{x+4} \, dx - (2 \times 6) \] Integrate \( \frac{36}{x+4} \):
\[ \int \frac{36}{x+4} \, dx = 36 \ln|x+4| \] Apply the limits from 0 to 2:
\[ [36 \ln|x+4|]_{0}^{2} - 12 \] \[ = (36 \ln|2+4|) - (36 \ln|0+4|) - 12 \] \[ = 36 \ln 6 - 36 \ln 4 - 12 \] Factor out 36:
\[ = 36 (\ln 6 - \ln 4) - 12 \] Using the logarithm property \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \):
\[ = 36 \ln \left(\frac{6}{4}\right) - 12 \] \[ = 36 \ln \left(\frac{3}{2}\right) - 12 \] Therefore, the consumer's surplus is \( 36 \ln \left(\frac{3}{2}\right) - 12 \) units. Logarithmic functions frequently appear in economic demand models.
In simple words: We first find how many items are bought when the price is Rs 6 using the given demand rule. Then, we use a special math tool (integration with logarithms) to figure out the total extra value customers get from buying at this price.

🎯 Exam Tip: When integrating functions like \( \frac{1}{ax+b} \), remember the result is \( \frac{1}{a} \ln|ax+b| \). Be mindful of log properties like \( \ln A - \ln B = \ln(A/B) \).

Question 8. The demand and supply functions under perfect competition are pd = 1600 – x² and ps = 2x² + 400 respectively, find the producer's surplus.
Answer:
Given the demand function \( p_d = 1600 - x^2 \) and supply function \( p_s = 2x^2 + 400 \).
Under perfect competition, equilibrium occurs when \( p_d = p_s \).
\( 1600 - x^2 = 2x^2 + 400 \)
Rearrange the terms to solve for \( x \):
\( 1600 - 400 = 2x^2 + x^2 \)
\( 1200 = 3x^2 \)
Divide by 3:
\( x^2 = \frac{1200}{3} \)
\( x^2 = 400 \)
Take the square root of both sides:
\( x = \pm\sqrt{400} \)
\( x = \pm 20 \)
Since quantity cannot be negative, we take the positive value:
\( x_0 = 20 \). This is the equilibrium quantity.
Now, substitute \( x_0 = 20 \) into either the demand or supply function to find the equilibrium price \( p_0 \). Using the demand function:
\( p_0 = 1600 - (20)^2 \)
\( p_0 = 1600 - 400 \)
\( p_0 = 1200 \). This is the equilibrium price.
The formula for Producer's Surplus (P.S) is:
\( P.S = (x_0 \times p_0) - \int_{0}^{x_0} g(x) \, dx \)
Substitute \( x_0 = 20 \), \( p_0 = 1200 \), and \( g(x) = 2x^2 + 400 \):
\[ P.S = (20 \times 1200) - \int_{0}^{20} (2x^2 + 400) \, dx \] \[ P.S = 24000 - \left[\frac{2x^3}{3} + 400x\right]_{0}^{20} \] First, evaluate the integral at the upper limit (20):
\[ \left[\frac{2(20)^3}{3} + 400(20)\right] = \left[\frac{2(8000)}{3} + 8000\right] \] \[ = \left[\frac{16000}{3} + 8000\right] \] Then, evaluate at the lower limit (0):
\[ \left[\frac{2(0)^3}{3} + 400(0)\right] = 0 \] Now, substitute these back into the P.S formula:
\[ P.S = 24000 - \left[\left(\frac{16000}{3} + 8000\right) - 0\right] \] \[ P.S = 24000 - \left(\frac{16000}{3} + \frac{24000}{3}\right) \] \[ P.S = 24000 - \frac{40000}{3} \] To subtract, find a common denominator for 24000:
\[ 24000 = \frac{24000 \times 3}{3} = \frac{72000}{3} \] \[ P.S = \frac{72000}{3} - \frac{40000}{3} \] \[ P.S = \frac{32000}{3} \] Therefore, the producer's surplus is \( \frac{32000}{3} \) units. This shows the additional benefit producers receive from selling their goods at the market price.
In simple words: First, we find the price and quantity where demand and supply are equal. Then, we calculate the total money sellers earn and subtract the minimum amount they would accept by using integration. This gives us the extra profit for the producers.

🎯 Exam Tip: When solving quadratic equations for quantity, always remember to consider only the positive real root, as quantity cannot be negative. Be meticulous with fraction arithmetic in the final steps.

Question 9. Under perfect competition for a commodity the demand and supply laws are Pd = \( \frac { 8 }{x+1} \) – 2 and Ps = \( \frac {x+3 }{2} \) respectively. Find the consumer's and producer's surplus.
Answer:
Given the demand function \( p_d = \frac{8}{x+1} - 2 \) and supply function \( p_s = \frac{x+3}{2} \).
Under perfect competition, equilibrium occurs when \( p_d = p_s \).
\( \frac{8}{x+1} - 2 = \frac{x+3}{2} \)
To solve this, first move the constant -2 to the right side:
\( \frac{8}{x+1} = \frac{x+3}{2} + 2 \)
Find a common denominator on the right side:
\( \frac{8}{x+1} = \frac{x+3}{2} + \frac{4}{2} \)
\( \frac{8}{x+1} = \frac{x+3+4}{2} \)
\( \frac{8}{x+1} = \frac{x+7}{2} \)
Cross-multiply:
\( 8 \times 2 = (x+7)(x+1) \)
\( 16 = x^2 + x + 7x + 7 \)
\( 16 = x^2 + 8x + 7 \)
Rearrange into a quadratic equation:
\( x^2 + 8x + 7 - 16 = 0 \)
\( x^2 + 8x - 9 = 0 \)
Factor the quadratic equation:
\( (x+9)(x-1) = 0 \)
This gives two possible values for \( x \): \( x = -9 \) or \( x = 1 \).
Since quantity cannot be negative, we choose \( x_0 = 1 \). This is the equilibrium quantity.
Now, substitute \( x_0 = 1 \) into either the demand or supply function to find the equilibrium price \( p_0 \). Using the supply function (it's simpler here):
\( p_0 = \frac{1+3}{2} \)
\( p_0 = \frac{4}{2} \)
\( p_0 = 2 \). This is the equilibrium price.

**Consumer's Surplus (C.S):**
\( C.S = \int_{0}^{x_0} f(x) \, dx - (p_0 \times x_0) \)
Substitute \( x_0 = 1 \), \( p_0 = 2 \), and \( f(x) = \frac{8}{x+1} - 2 \):
\[ C.S = \int_{0}^{1} \left(\frac{8}{x+1} - 2\right) \, dx - (1 \times 2) \] Integrate \( \left(\frac{8}{x+1} - 2\right) \):
\[ \int \left(\frac{8}{x+1} - 2\right) \, dx = 8 \ln|x+1| - 2x \] Apply the limits from 0 to 1:
\[ [8 \ln|x+1| - 2x]_{0}^{1} - 2 \] \[ = [8 \ln(1+1) - 2(1)] - [8 \ln(0+1) - 2(0)] - 2 \] \[ = [8 \ln 2 - 2] - [8 \ln 1 - 0] - 2 \] Since \( \ln 1 = 0 \):
\[ = [8 \ln 2 - 2] - [0] - 2 \] \[ = 8 \ln 2 - 2 - 2 \] \[ C.S = 8 \ln 2 - 4 \] Therefore, the consumer's surplus is \( (8 \ln 2 - 4) \) units.

**Producer's Surplus (P.S):**
\( P.S = (x_0 \times p_0) - \int_{0}^{x_0} g(x) \, dx \)
Substitute \( x_0 = 1 \), \( p_0 = 2 \), and \( g(x) = \frac{x+3}{2} \):
\[ P.S = (1 \times 2) - \int_{0}^{1} \frac{x+3}{2} \, dx \] \[ P.S = 2 - \frac{1}{2} \int_{0}^{1} (x+3) \, dx \] Integrate \( (x+3) \):
\[ \int (x+3) \, dx = \frac{x^2}{2} + 3x \] Apply the limits from 0 to 1:
\[ P.S = 2 - \frac{1}{2} \left[\frac{x^2}{2} + 3x\right]_{0}^{1} \] \[ P.S = 2 - \frac{1}{2} \left[\left(\frac{(1)^2}{2} + 3(1)\right) - \left(\frac{(0)^2}{2} + 3(0)\right)\right] \] \[ P.S = 2 - \frac{1}{2} \left[\left(\frac{1}{2} + 3\right) - 0\right] \] \[ P.S = 2 - \frac{1}{2} \left[\frac{1+6}{2}\right] \] \[ P.S = 2 - \frac{1}{2} \left[\frac{7}{2}\right] \] \[ P.S = 2 - \frac{7}{4} \] To subtract, find a common denominator:
\[ P.S = \frac{8}{4} - \frac{7}{4} \] \[ P.S = \frac{1}{4} \] Therefore, the producer's surplus is \( \frac{1}{4} \) units. These calculations show the benefits accrued to both consumers and producers in a perfectly competitive market.
In simple words: First, we find the meeting point of supply and demand to get the market price and quantity. Then, we use integration to calculate the extra value buyers get (consumer surplus) and the extra profit sellers make (producer surplus) at that market point.

🎯 Exam Tip: Be very careful when solving for equilibrium quantity with rational functions, as it often leads to quadratic equations. Remember to exclude negative quantities. Also, clearly distinguish between the demand and supply functions when applying the surplus formulas.

Question 10. The demand equation for a products is x = \( \sqrt {100-p} \) and the supply equation is x = \( \frac{p}{2} - 10 \). Determine the consumer's surplus and producer's, under market equilibrium.
Answer:
Given the demand equation \( x_d = \sqrt{100-p} \) and the supply equation \( x_s = \frac{p}{2} - 10 \).
Under market equilibrium, \( x_d = x_s \).
\( \sqrt{100-p} = \frac{p}{2} - 10 \)
To solve for \( p \), square both sides of the equation:
\( (\sqrt{100-p})^2 = \left(\frac{p}{2} - 10\right)^2 \)
\( 100 - p = \left(\frac{p}{2}\right)^2 - 2\left(\frac{p}{2}\right)(10) + (10)^2 \)
\( 100 - p = \frac{p^2}{4} - 10p + 100 \)
To eliminate the fraction, multiply the entire equation by 4:
\( 4(100 - p) = 4\left(\frac{p^2}{4}\right) - 4(10p) + 4(100) \)
\( 400 - 4p = p^2 - 40p + 400 \)
Move all terms to one side to form a quadratic equation:
\( 0 = p^2 - 40p + 4p + 400 - 400 \)
\( 0 = p^2 - 36p \)
Factor out \( p \):
\( p(p - 36) = 0 \)
This gives two possible values for \( p \): \( p = 0 \) or \( p = 36 \).
If \( p = 0 \), then from the supply function \( x = \frac{0}{2} - 10 = -10 \), which is not possible (quantity cannot be negative).
So, the equilibrium price \( p_0 = 36 \).
Now, substitute \( p_0 = 36 \) into either the demand or supply equation to find the equilibrium quantity \( x_0 \). Using the demand equation:
\( x_0 = \sqrt{100 - 36} \)
\( x_0 = \sqrt{64} \)
\( x_0 = 8 \). This is the equilibrium quantity.

**Consumer's Surplus (C.S):**
For consumer's surplus, we need the demand function \( p = f(x) \). From \( x = \sqrt{100-p} \):
\( x^2 = 100 - p \)
\( p = 100 - x^2 \). This is the inverse demand function, \( f(x) \).
\( C.S = \int_{0}^{x_0} f(x) \, dx - (p_0 \times x_0) \)
Substitute \( x_0 = 8 \), \( p_0 = 36 \), and \( f(x) = 100 - x^2 \):
\[ C.S = \int_{0}^{8} (100 - x^2) \, dx - (36 \times 8) \] \[ C.S = \left[100x - \frac{x^3}{3}\right]_{0}^{8} - 288 \] Apply the limits from 0 to 8:
\[ C.S = \left[100(8) - \frac{(8)^3}{3}\right] - \left[100(0) - \frac{(0)^3}{3}\right] - 288 \] \[ C.S = \left[800 - \frac{512}{3}\right] - 0 - 288 \] \[ C.S = 800 - \frac{512}{3} - 288 \] \[ C.S = 512 - \frac{512}{3} \] Factor out 512:
\[ C.S = 512 \left(1 - \frac{1}{3}\right) \] \[ C.S = 512 \left(\frac{3-1}{3}\right) \] \[ C.S = 512 \left(\frac{2}{3}\right) \] \[ C.S = \frac{1024}{3} \] Therefore, the consumer's surplus is \( \frac{1024}{3} \) units.

**Producer's Surplus (P.S):**
For producer's surplus, we need the supply function \( p = g(x) \). From \( x = \frac{p}{2} - 10 \):
\( x + 10 = \frac{p}{2} \)
\( p = 2(x + 10) \)
\( p = 2x + 20 \). This is the inverse supply function, \( g(x) \).
\( P.S = (x_0 \times p_0) - \int_{0}^{x_0} g(x) \, dx \)
Substitute \( x_0 = 8 \), \( p_0 = 36 \), and \( g(x) = 2x + 20 \):
\[ P.S = (8 \times 36) - \int_{0}^{8} (2x + 20) \, dx \] \[ P.S = 288 - \left[\frac{2x^2}{2} + 20x\right]_{0}^{8} \] \[ P.S = 288 - \left[x^2 + 20x\right]_{0}^{8} \] Apply the limits from 0 to 8:
\[ P.S = 288 - \left[(8)^2 + 20(8)\right] - \left[(0)^2 + 20(0)\right] \] \[ P.S = 288 - [64 + 160] - 0 \] \[ P.S = 288 - 224 \] \[ P.S = 64 \] Therefore, the producer's surplus is 64 units. Understanding both surpluses helps in analyzing market efficiency and welfare.
In simple words: First, we find the equilibrium price and quantity by setting the demand and supply equations equal. Then, we use special math tools (integration) with the transformed demand and supply rules to find the extra value buyers get and the extra profit sellers make.

🎯 Exam Tip: When demand or supply is given as \( x \) in terms of \( p \), remember to convert them to \( p \) in terms of \( x \) (inverse demand/supply functions) before applying the integration formulas for consumer's and producer's surplus. Always check for valid economic solutions (e.g., non-negative price and quantity).

Question 11. Find the consumer's surplus and producer's surplus for the demand function pd = 25 – 3x and supply function ps = 5 + 2x
Answer:
Given the demand function \( p_d = 25 - 3x \) and supply function \( p_s = 5 + 2x \).
Under market equilibrium, \( p_d = p_s \).
\( 25 - 3x = 5 + 2x \)
Rearrange the terms to solve for \( x \):
\( 25 - 5 = 2x + 3x \)
\( 20 = 5x \)
Divide by 5:
\( x = \frac{20}{5} \)
\( x = 4 \)
So, the equilibrium quantity \( x_0 = 4 \).
Now, substitute \( x_0 = 4 \) into either the demand or supply function to find the equilibrium price \( p_0 \). Using the demand function:
\( p_0 = 25 - 3(4) \)
\( p_0 = 25 - 12 \)
\( p_0 = 13 \).
Thus, the equilibrium quantity is 4 units and the equilibrium price is Rs 13.

**Consumer's Surplus (C.S):**
\( C.S = \int_{0}^{x_0} f(x) \, dx - (p_0 \times x_0) \)
Substitute \( x_0 = 4 \), \( p_0 = 13 \), and \( f(x) = 25 - 3x \):
\[ C.S = \int_{0}^{4} (25 - 3x) \, dx - (4 \times 13) \] \[ C.S = \left[25x - \frac{3x^2}{2}\right]_{0}^{4} - 52 \] Apply the limits from 0 to 4:
\[ C.S = \left[25(4) - \frac{3(4)^2}{2}\right] - \left[25(0) - \frac{3(0)^2}{2}\right] - 52 \] \[ C.S = \left[100 - \frac{3(16)}{2}\right] - 0 - 52 \] \[ C.S = [100 - 3(8)] - 52 \] \[ C.S = [100 - 24] - 52 \] \[ C.S = 76 - 52 \] \[ C.S = 24 \] Therefore, the consumer's surplus is 24 units.

**Producer's Surplus (P.S):**
\( P.S = (x_0 \times p_0) - \int_{0}^{x_0} g(x) \, dx \)
Substitute \( x_0 = 4 \), \( p_0 = 13 \), and \( g(x) = 5 + 2x \):
\[ P.S = (4 \times 13) - \int_{0}^{4} (5 + 2x) \, dx \] \[ P.S = 52 - \left[5x + \frac{2x^2}{2}\right]_{0}^{4} \] \[ P.S = 52 - \left[5x + x^2\right]_{0}^{4} \] Apply the limits from 0 to 4:
\[ P.S = 52 - \left[(5(4) + (4)^2) - (5(0) + (0)^2)\right] \] \[ P.S = 52 - [(20 + 16) - 0] \] \[ P.S = 52 - 36 \] \[ P.S = 16 \] Therefore, the producer's surplus is 16 units. Calculating both surpluses provides a complete picture of market benefits.
In simple words: First, we find the market price and quantity where demand and supply meet. Then, using integration, we calculate the extra value customers gain (consumer surplus) and the extra profit sellers make (producer surplus) at this market point.

🎯 Exam Tip: Clearly identify and use the correct function (demand for consumer surplus, supply for producer surplus) in the integral. Ensure you accurately calculate the equilibrium price and quantity, as errors there will affect both surplus calculations.