GSEB Class 9 Maths Solutions Chapter 13 Surface Areas and Volumes Exercise 13.3

Question 1. The diameter of the base of the cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area.
Answer: The base diameter is \( 10.5 \) cm. So, the base radius \( (r) \) is \( \frac{10.5}{2} \) cm, which equals \( 5.25 \) cm. The slant height \( (l) \) is \( 10 \) cm. To find the curved surface area of the cone, we use the formula \( \pi rl \). Substituting the values, we get \( \frac{22}{7} \times 5.25 \times 10 = 165 \) cm².
In simple words: First, halve the diameter to get the radius. Then, use the formula for curved surface area of a cone, which is pi times radius times slant height, to get the final area.

Exam Tip: Remember to correctly identify the given values for radius, height, and slant height. Ensure you use the correct formula for curved surface area, which is \( \pi rl \).

Question 2. Find the total surface area of a cone, if its slant height is 21 m and the diameter of its base is 24 m.
Answer: The slant height \( (l) \) is \( 21 \) m. The base diameter is \( 24 \) m, so the base radius \( (r) \) is \( \frac{24}{2} \) m, which comes to \( 12 \) m. The total surface area of a cone is found using the formula \( \pi r (l + r) \). Substituting the values, we get \( \frac{22}{7} \times 12 \times (21 + 12) = \frac{22}{7} \times 12 \times 33 = \frac{8712}{7} = 1244 \frac{4}{7} \) m².
In simple words: Calculate the radius from the diameter. Then, use the total surface area formula for a cone, which is pi times radius times (slant height plus radius).

Exam Tip: Distinguish between curved surface area \( (\pi rl) \) and total surface area \( (\pi r(l+r)) \). Always ensure you are calculating the correct area asked in the question.

Question 3. The curved surface area of a cone is 308 cm2 and its slant height is 14 cm. Find 1. the radius of the base and 2. the total surface area of the cone.
Answer:
1. The slant height \( (l) \) is \( 14 \) cm and the curved surface area is \( 308 \) cm². We know the formula for curved surface area is \( \pi rl \). So, \( \frac{22}{7} \times r \times 14 = 308 \). Solving for \( r \), we get \( r = \frac{308 \times 7}{22 \times 14} = 7 \) cm. Therefore, the base radius is \( 7 \) cm.
2. The total surface area of the cone is calculated using \( \pi r (l + r) \). Using the found radius and given slant height, we have \( \frac{22}{7} \times 7 \times (14 + 7) = \frac{22}{7} \times 7 \times 21 = 462 \) cm². So, the cone's total surface area is \( 462 \) cm².
In simple words: First, use the curved surface area formula to work out the radius. Then, use this radius along with the slant height in the total surface area formula to find the second part of the answer.

Exam Tip: When a question asks for multiple parts, address each part separately and clearly. Double-check your calculations, especially when using a value derived from a previous step.

Question 4. A conical tent is 10 m high and the radius of its base is 24 m. Find: 1. the slant height of the tent 2. cost of the canvas required to make the tent, if the cost of 1 m² canvas is Rs. 70.
Answer:
1. The height \( (h) \) of the tent is \( 10 \) m and its radius \( (r) \) is \( 24 \) m. The slant height \( (l) \) can be found using the formula \( l = \sqrt{r^2 + h^2} \). So, \( l = \sqrt{(24)^2 + (10)^2} = \sqrt{576 + 100} = \sqrt{676} = 26 \) m. Thus, the tent's slant height is \( 26 \) m.
2. The curved surface area of the tent is \( \pi rl \). Substituting the values, we get \( \frac{22}{7} \times 24 \times 26 \) m². If the canvas costs Rs. \( 70 \) per m², the total cost will be \( \frac{22}{7} \times 24 \times 26 \times 70 = Rs. 1,37,280 \). Therefore, the canvas will cost Rs. \( 1,37,280 \).
In simple words: First, find the slant height using the Pythagorean theorem with height and radius. Then, calculate the curved surface area of the tent. Finally, multiply this area by the cost per square meter to get the total cost.

Exam Tip: For problems involving conical tents, the canvas needed covers the curved surface area only, not the base. Always use \( \pi = \frac{22}{7} \) unless specified otherwise.

Question 5. What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base radius 6 m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20 cm. (Use \( \pi = 3.14 \))
Answer: For a conical tent, the height \( (h) \) is \( 8 \) m and the base radius \( (r) \) is \( 6 \) m. First, we find the slant height \( (l) \) using \( l = \sqrt{r^2 + h^2} = \sqrt{(6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \) m. The curved surface area, which is the amount of tarpaulin needed, is \( \pi rl = 3.14 \times 6 \times 10 = 188.4 \) m². Since the tarpaulin width is \( 3 \) m, its length will be \( \frac{188.4}{3} = 62.8 \) m. An extra \( 20 \) cm, or \( 0.2 \) m, is needed for stitching and waste. So, the actual required length of tarpaulin is \( 62.8 + 0.2 = 63 \) m.
In simple words: Figure out the tent's slant height, then its curved surface area. Divide this area by the tarpaulin's width to get the needed length. Finally, add the extra length for seams and cutting.

Exam Tip: Pay attention to units, converting cm to m when necessary. Always add any extra material required for overlaps or wastage to the calculated length.

Question 6. The slant height and a base diameter of a conical tomb is 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of Rs. 210 per 100 m².
Answer: The slant height \( (l) \) of the conical tomb is \( 25 \) m, and its base diameter \( (d) \) is \( 14 \) m. Thus, the base radius \( (r) \) is \( \frac{14}{2} = 7 \) m. The curved surface area of the tomb is \( \pi rl = \frac{22}{7} \times 7 \times 25 = 550 \) m². The white-washing cost is Rs. \( 210 \) for every \( 100 \) m². So, for \( 550 \) m², the total cost will be \( \frac{210}{100} \times 550 = Rs. 1155 \).
In simple words: Calculate the radius from the diameter. Then, find the curved surface area of the tomb. Finally, multiply this area by the cost rate per square meter to get the total whitewashing expense.

Exam Tip: Remember to calculate the cost based on the curved surface area, as white-washing is typically applied to the exposed external surface. Pay close attention to the rate given (e.g., per 100 m²).

Question 7. A joker's cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.
Answer: For a joker's cap, the base radius \( (r) \) is \( 7 \) cm and the height \( (h) \) is \( 24 \) cm. First, we determine the slant height \( (l) \) using \( l = \sqrt{r^2 + h^2} = \sqrt{(7)^2 + (24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \) cm. The curved surface area of one cap is \( \pi rl = \frac{22}{7} \times 7 \times 25 = 550 \) cm². To make \( 10 \) such caps, the total sheet area needed will be \( 550 \times 10 = 5500 \) cm².
In simple words: First, find the slant height of one cap using the given radius and height. Then, calculate the curved surface area for a single cap. Multiply this area by 10 to get the total sheet needed for all caps.

Exam Tip: For conical caps, only the curved surface area is relevant. Remember to calculate the total area if multiple identical items are requested.

Question 8. A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and a height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is Rs. 12 per m², what will be the cost of painting all these cones? (Use \( \pi = 3.14 \) and take \( \sqrt{1.04} = 1.02 \))
Answer: Each cone has a base diameter of \( 40 \) cm, which means its base radius \( (r) \) is \( \frac{40}{2} = 20 \) cm, or \( 0.2 \) m. The height \( (h) \) of each cone is \( 1 \) m. We find the slant height \( (l) \) using \( l = \sqrt{r^2 + h^2} = \sqrt{(0.2)^2 + (1)^2} = \sqrt{0.04 + 1} = \sqrt{1.04} = 1.02 \) m. The curved surface area of one cone is \( \pi rl = 3.14 \times 0.2 \times 1.02 = 0.64056 \) m². For \( 50 \) cones, the total curved surface area is \( 50 \times 0.64056 = 32.028 \) m². If painting costs Rs. \( 12 \) per m², the total cost for all cones will be \( 32.028 \times 12 = Rs. 384.336 \), which is approximately Rs. \( 384.34 \).
In simple words: Convert all measurements to meters. Calculate the slant height of one cone. Find the curved surface area of a single cone, then multiply by the number of cones. Finally, multiply this total area by the painting cost per square meter to get the full expense.

Exam Tip: Ensure all units are consistent (e.g., convert cm to m at the beginning). When dealing with multiple objects, remember to multiply the single-object area by the total count before calculating the final cost.