Question 1: Calculate the surface area of a sphere with radius 5 cm using the formula 4πr². Show your working.

Answer: 314.16

Question 2: A sphere has a radius of 8 cm. What is its surface area? Use 3.14 for π. Show your working.

Answer: 804.2

Question 3: Calculate the surface area of a sphere with radius 12 mm. Use 3.14 for π. Show your working.

Answer: 1805.76

Question 4: If the radius of a sphere is doubled from 3 cm to 6 cm, by what factor does the surface area increase? Explain your reasoning.

Answer: The surface area increases by a factor of 4 because (2r)² = 4r², so 4π(2r)²=4×4πr²

Question 1: A sphere has a surface area of 125.6 cm². Calculate its radius, giving your answer to two decimal places. Use π≈3.14.

Answer: r=5.00

Question 2: A spherical balloon has a radius of 10 inches. If the surface area is mistakenly calculated using 3 instead of 3.14 for π, what is the error in the surface area calculation? Show the incorrect and correct answers.

Answer: Incorrect: 4×3×10²=1200; Correct: 4×3.14×10²=1256

Question 3: Explain why using an approximation of π as 3 instead of 3.14 leads to an underestimation of the surface area. Support your explanation with calculations.

Answer: Using π=3 instead of 3.14 underestimates the surface area because 4×3×r²<4×3.14×r², leading to a smaller calculated value.

Question 1: A sports ball has a radius of 9 cm. Calculate the surface area to determine how much material is needed to cover it entirely.

Answer: 4×3.14×9²=1017.88 cm²

Question 2: A spherical water tank has a radius of 2 meters. If a coating is applied to cover the entire surface, what is the area of the coating needed? Use π=3.14.

Answer: 4×3.14×2²=50.24 m²

Question 1: A sphere's surface area is increased from 200π cm² to 800π cm². Find the original and new radii, explaining your reasoning.

Answer: Original r=5cm, New r=10cm

Question 2: Derive the formula for the surface area of a sphere starting from the volume formula. Explain each step.

Answer: This is a conceptual derivation; students should explain the relation between volume and surface area, involving integration or known formulas.

Question 1: A student calculates the surface area of a sphere with radius 7 cm as 4×22/7×7². Identify the mistake and correct the calculation.

Answer: Using 22/7 for π gives 4×22/7×49=608.57; correct: 4×3.14×49=615.44

Question 2: A common misconception is that doubling the radius doubles the surface area. Is this true? Explain with calculation.

Answer: No, surface area increases by a factor of four because (2r)²=4r².

Question 3: Identify and correct the mistake: A student calculates surface area as 4πr³.

Answer: Incorrect formula; should be 4πr², not 4πr³.