If the radius of a sphere is increased by $ 20\%$, then the new volume will become ________ of the original vo

Question

If the radius of a sphere is increased by $ 20\%$, then the new volume will become ________ of the original volume.

Examoogle · Accepted Answer

The correct answer is Option C: 172.8%. Use the successsive percentage formula for 3 dimensions (n=3): (1 + r/100)^3 - 1. For 20%, (1.2)^3 = 1.728, which is 172.8%.

Step-by-Step Solution & Concepts:
Shortcut Trick: Use the successsive percentage formula for 3 dimensions (n=3): (1 + r/100)^3 - 1. For 20%, (1.2)^3 = 1.728, which is 172.8%.

Formula Used: $$V_{new} = \frac{4}{3}\pi (R_{new})^3$$

Given: Initial radius of a sphere = R, percentage increase in radius = 20%.

Steps: Define original volume: Let the original radius be $R$. The volume of a sphere is given by $V = \frac{4}{3}\pi R^3$.. Calculate new radius: Increasing the radius by 20% means the new radius $R_{new} = R + 0.20R = 1.2R$.. Calculate new volume: Substitute $R_{new}$ into the volume formula: $V_{new} = \frac{4}{3}\pi (1.2R)^3$.. Express as percentage: The new volume is $1.728$ times the original volume $V$. Converting to percentage, $1.728 	imes 100\% = 172.8\%$.

Answer: C is correct because the new volume becomes 172.8% of the original volume after increasing the radius by 20%.

Exam Tip: This concept of scaling factor $k^3$ for volume (where $k$ is the linear scale factor) applies to all 3D shapes, including cylinders and cubes.

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