Question

A boy, Chintu while playing with modelling clay, changes a cone shaped toy into a spherical toy. The height of the cone was 20cm and radius of base was 5 cm. Find the radius and the surface area of the sphere

• A. 308 sqcm
• B. 310sqcm
• C. 312sqcm
• D. 314sqcm

Hint:

When a solid is melted and recast into another solid, the volume remains constant. Remember the formulas for volumes and surface areas of common 3D shapes.

Answer 1

The Correct Option is D

When a cone shaped toy is changed into a spherical toy using the same modelling clay, the volume of the clay remains constant. Thus, the volume of the cone is equal to the volume of the sphere. Given:
Height of the cone, \( h = 20 \) cm
Radius of the base of the cone, \( r_{cone} = 5 \) cm
Step 1: Calculate the volume of the cone.
The formula for the volume of a cone is \( V_{cone} = \frac{1}{3}\pi r_{cone}^2 h \).
Substituting the given values: \[ V_{cone} = \frac{1}{3} \times \pi \times (5)^2 \times 20 \] \[ V_{cone} = \frac{1}{3} \times \pi \times 25 \times 20 \] \[ V_{cone} = \frac{500\pi}{3} \text{ cm}^3 \] Step 2: Equate the volume of the cone to the volume of the sphere and find the radius of the sphere.
The formula for the volume of a sphere is \( V_{sphere} = \frac{4}{3}\pi r_{sphere}^3 \), where \( r_{sphere} \) is the radius of the sphere.
Since \( V_{cone} = V_{sphere} \): \[ \frac{500\pi}{3} = \frac{4}{3}\pi r_{sphere}^3 \] Cancel out \( \frac{\pi}{3} \) from both sides: \[ 500 = 4 r_{sphere}^3 \] Divide by 4: \[ r_{sphere}^3 = \frac{500}{4} \] \[ r_{sphere}^3 = 125 \] Take the cube root of both sides to find \( r_{sphere} \): \[ r_{sphere} = \sqrt[3]{125} \] \[ r_{sphere} = 5 \text{ cm} \] So, the radius of the sphere is 5 cm. Step 3: Calculate the surface area of the sphere.
The formula for the surface area of a sphere is \( SA_{sphere} = 4\pi r_{sphere}^2 \).
Substituting the value of \( r_{sphere} = 5 \) cm: \[ SA_{sphere} = 4 \times \pi \times (5)^2 \] \[ SA_{sphere} = 4 \times \pi \times 25 \] \[ SA_{sphere} = 100\pi \text{ cm}^2 \] Using the approximate value \( \pi \approx 3.14 \) (as indicated by the options): \[ SA_{sphere} = 100 \times 3.14 \] \[ SA_{sphere} = 314 \text{ cm}^2 \] Therefore, the surface area of the sphere is 314 sqcm.