Exercise 11.1 Solutions (Class 9)
Surface Areas and Volumes
(Assume π = 22/7, unless stated otherwise)
1. Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area.
Given: Diameter = 10.5 cm, Slant height (l) = 10 cm.
Step 1: Find the radius (r).
Radius (r) = Diameter / 2 = 10.5 / 2 = 5.25 cm.
Step 2: Calculate the Curved Surface Area (CSA).
CSA of a cone = πrl
= (22/7) × 5.25 × 10
= (22/7) × 52.5 = 22 × 7.5 = 165 cm².
Answer: The curved surface area of the cone is 165 cm².
2. Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
Given: Slant height (l) = 21 m, Diameter = 24 m.
Step 1: Find the radius (r).
Radius (r) = Diameter / 2 = 24 / 2 = 12 m.
Step 2: Calculate the Total Surface Area (TSA).
TSA of a cone = πr(l + r)
= (22/7) × 12 × (21 + 12)
= (22/7) × 12 × 33 = 8712 / 7 ≈ 1244.57 m².
Answer: The total surface area of the cone is approximately 1244.57 m².
3. Curved surface area of a cone is 308 cm² and its slant height is 14 cm. Find (i) radius of the base and (ii) total surface area of the cone.
Given: CSA = 308 cm², Slant height (l) = 14 cm.
(i) Find the radius of the base (r).
CSA = πrl => 308 = (22/7) × r × 14
308 = 44 × r
r = 308 / 44 = 7 cm.
(ii) Find the total surface area of the cone (TSA).
TSA = CSA + Area of base = CSA + πr²
= 308 + (22/7) × 7² = 308 + 154 = 462 cm².
Answer: Radius is 7 cm and Total Surface Area is 462 cm².
4. A conical tent is 10 m high and the radius of its base is 24 m. Find (i) slant height of the tent. (ii) cost of the canvas required to make the tent, if the cost of 1 m² canvas is ₹70.
Given: Height (h) = 10 m, Radius (r) = 24 m.
(i) Find the slant height (l).
l = √(r² + h²) = √(24² + 10²) = √(576 + 100) = √676 = 26 m.
(ii) Find the cost of the canvas.
The canvas required is the Curved Surface Area (CSA).
CSA = πrl = (22/7) × 24 × 26 = 13728 / 7 m².
Cost = Area × Rate = (13728 / 7) × 70 = 13728 × 10 = ₹137,280.
Answer: Slant height is 26 m and the cost is ₹1,37,280.
5. What length of tarpaulin 3 m wide will be required to make a 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 π = 3.14).
Given: Height (h) = 8 m, Radius (r) = 6 m, Width of tarpaulin = 3 m.
Step 1: Find the slant height (l) of the tent.
l = √(r² + h²) = √(6² + 8²) = √(36 + 64) = √100 = 10 m.
Step 2: Find the area of tarpaulin required (CSA).
Area = πrl = 3.14 × 6 × 10 = 188.4 m².
Step 3: Find the length of the tarpaulin roll.
Length × Width = Area => Length × 3 = 188.4 => Length = 188.4 / 3 = 62.8 m.
Step 4: Add the extra length for wastage.
Extra length = 20 cm = 0.2 m.
Total Length = 62.8 + 0.2 = 63 m.
Answer: The required length of tarpaulin is 63 m.
6. The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of ₹210 per 100 m².
Given: Slant height (l) = 25 m, Diameter = 14 m.
Step 1: Find the radius (r) and CSA.
Radius (r) = 14 / 2 = 7 m.
CSA = πrl = (22/7) × 7 × 25 = 550 m².
Step 2: Calculate the cost of white-washing.
Cost per 1 m² = 210 / 100 = ₹2.10.
Total Cost = Area × Rate = 550 × 2.10 = ₹1155.
Answer: The cost of white-washing is ₹1155.
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.
Given: Radius (r) = 7 cm, Height (h) = 24 cm.
Step 1: Find the slant height (l).
l = √(r² + h²) = √(7² + 24²) = √(49 + 576) = √625 = 25 cm.
Step 2: Find the area of the sheet for one cap (CSA).
Area = πrl = (22/7) × 7 × 25 = 550 cm².
Step 3: Find the total area for 10 caps.
Total Area = 10 × 550 = 5500 cm².
Answer: The area of the sheet required is 5500 cm².
8. A bus stop is barricaded… by using 50 hollow cones… If the outer side of each of the cones is to be painted and the cost of painting is ₹12 per m², what will be the cost of painting all these cones? (Use π = 3.14 and take √1.04 = 1.02)
Given: Diameter = 40 cm = 0.4 m, Height (h) = 1 m.
Step 1: Find the radius (r) and slant height (l) of one cone.
Radius (r) = 0.4 / 2 = 0.2 m.
l = √(r² + h²) = √((0.2)² + 1²) = √(0.04 + 1) = √1.04 = 1.02 m.
Step 2: Find the CSA of one cone.
CSA = πrl = 3.14 × 0.2 × 1.02 = 0.64056 m².
Step 3: Find the total area to be painted (for 50 cones).
Total Area = 50 × 0.64056 = 32.028 m².
Step 4: Calculate the total cost of painting.
Total Cost = Total Area × Rate = 32.028 × 12 = ₹384.336.
Answer: The cost of painting all the cones is approximately ₹384.34.
