A sphere has the same curved surface area as a right circular cone with vertical height 40 cm and base radius 30 cm. What is the radius of the sphere in centimetres?

Introduction / Context:
This question checks your understanding of curved surface areas of common 3D solids, specifically a right circular cone and a sphere. In many aptitude and engineering entrance exams, you are asked to relate areas or volumes of different solids that are melted or equated. Recognising the correct formulas and equating them is the key to answering quickly and correctly.

Given Data / Assumptions:
• A right circular cone has vertical height h = 40 cm. • Radius of the cone's base r = 30 cm. • A sphere has the same curved surface area as this cone. • We are asked to find the radius R of the sphere.

Concept / Approach:
The curved surface area of a right circular cone is given by CSA_cone = π * r * l, where r is the base radius and l is the slant height. The curved surface area of a sphere is given by CSA_sphere = 4 * π * R^2, where R is the radius of the sphere. Since the question states that these curved surface areas are equal, we equate the two formulas and solve for R. First, we must compute the slant height l of the cone using the Pythagoras theorem in the right triangle formed by radius, height and slant height.

Step-by-Step Solution:
1. Height of the cone h = 40 cm and radius r = 30 cm. 2. Find the slant height l of the cone: l = sqrt(h^2 + r^2). 3. Compute h^2 = 40^2 = 1600 and r^2 = 30^2 = 900. 4. Then l = sqrt(1600 + 900) = sqrt(2500) = 50 cm. 5. Curved surface area of cone: CSA_cone = π * r * l = π * 30 * 50 = 1500π. 6. Curved surface area of sphere: CSA_sphere = 4 * π * R^2. 7. Set CSA_cone = CSA_sphere: 4 * π * R^2 = 1500π. 8. Cancel π on both sides to get 4 * R^2 = 1500. 9. So R^2 = 1500 / 4 = 375. 10. Write 375 as 25 * 15, so R = sqrt(25 * 15) = 5 * sqrt(15).

Verification / Alternative check:
You can verify by substituting R = 5 * sqrt(15) back into the sphere's curved surface area formula. Then CSA_sphere = 4 * π * (5 * sqrt(15))^2 = 4 * π * 25 * 15 = 1500π, which exactly matches the cone's curved surface area. This confirms that the derived radius is consistent. Also, note that the radius is slightly less than 5 * 4 = 20 cm, which is reasonable compared to the cone dimensions.

Why Other Options Are Wrong:
• 5√5 cm: This would give R^2 = 25 * 5 = 125, and CSA_sphere = 4 * π * 125 = 500π, which is too small. • 5√3 cm: Here R^2 = 75, so CSA_sphere = 300π, still much less than 1500π. • 5√10 cm: This gives R^2 = 250, so CSA_sphere = 1000π, which is still less than 1500π. • 10 cm: Then CSA_sphere = 4 * π * 100 = 400π, again far from 1500π.

Common Pitfalls:
A frequent mistake is to equate the volumes instead of the curved surface areas, which completely changes the equation. Another common error is to forget to compute or correctly use the slant height of the cone, leading to wrong values of area. Some students also forget to cancel π from both sides and end up carrying unnecessary factors through the calculation. Staying organised with formulas and algebraic steps avoids these errors.

Final Answer:
The radius of the sphere is 5√15 cm.