Question 1: Calculate the surface area of a cone with a radius of 3 cm and a slant height of 5 cm. Use π ≈ 3.14.

Answer: πr² + πrl = 3.14×3² + 3.14×3×5 = 28.26 + 47.1 = 75.36 cm²

Question 2: A cone has a radius of 4 cm and a slant height of 6 cm. Find its surface area, rounding to two decimal places.

Answer: πr² + πrl = 3.14×4² + 3.14×4×6 = 50.24 + 75.36 = 125.60 cm²

Question 3: If a cone's surface area is 150 cm² and the slant height is 7 cm, what is the radius? (Use π ≈ 3.14)

Answer: r = sqrt((150 - πrl)/π) = sqrt((150 - 3.14×r×7)/3.14)

Question 4: Construct a cone with a radius of 2.5 cm and a slant height of 4 cm. Calculate its surface area.

Answer: πr² + πrl = 3.14×2.5² + 3.14×2.5×4 = 19.625 + 31.4 = 51.02 cm²

Question 5: Calculate the lateral surface area (πrl) of a cone with radius 6 cm and slant height 10 cm.

Answer: πrl = 3.14×6×10 = 188.4 cm²

Question 6: A cone has a surface area of 200 cm² and a radius of 4 cm. Find its slant height. (Use π ≈ 3.14)

Answer: Determine l from 200 = 3.14×4² + 3.14×4×l → 200 = 50.24 + 12.56l → 12.56l = 149.76 → l ≈ 11.93 cm

Question 7: A student claims the surface area of a cone with radius 3 cm and slant height 5 cm is 70 cm². Identify and correct the mistake in their calculation.

Answer: Incorrect calculation: surface area = πr² + πrl = 28.26 + 47.1 = 75.36 cm², not 70 cm²

Question 8: A cone's total surface area is 100 cm². If the radius is 2 cm, find the slant height. Use π ≈ 3.14.

Answer: 100 = 3.14×2² + 3.14×2×l → 100 = 12.56 + 6.28l → 6.28l = 87.44 → l ≈ 13.92 cm

Question 9: A real-world scenario: A toy cone has a radius of 5 cm and a slant height of 8 cm. Calculate its surface area to determine the amount of wrapping paper needed.

Answer: πr² + πrl = 3.14×25 + 3.14×5×8 = 78.5 + 125.6 = 204.10 cm²

Question 10: Challenge: Derive the surface area formula for a cone starting from the lateral area and base area. Explain each step clearly.

Answer: The total surface area is the sum of the base area πr² and the lateral area πrl, which accounts for the cone's curved surface. Thus, total SA = πr² + πrl.

Question 11: An extension problem: If the surface area of a cone is doubled, and the radius remains the same, how does the slant height change? Express your answer in terms of the original slant height.

Answer: Original SA: S = πr² + πrl; doubled SA: 2S = πr² + πrL' → L' = 2l - (πr²/πr) = 2l - r