Question

If the radius of a right circular cone is increased by 50%, its volume increases by

• A. 75%
• B. 100%
• C. 125%
• D. 237.5%

Answer 1

The Correct Option is C

The volume \( V \) of a right circular cone is given by the formula:

\[ V = \frac{1}{3} \pi r^2 h \]

where \( r \) is the radius and \( h \) is the height. If the radius is increased by 50%, the new radius \( r' \) becomes:

\[ r' = 1.5r \]

Substituting \( r' \) into the volume formula, the new volume \( V' \) is:

\[ V' = \frac{1}{3} \pi (1.5r)^2 h \]

Simplifying:

\[ V' = \frac{1}{3} \pi (2.25r^2) h \]

\[ V' = 2.25 \cdot \frac{1}{3} \pi r^2 h \]

\[ V' = 2.25V \]

The increase in volume is:

\[ \Delta V = V' - V = 2.25V - V = 1.25V \]

To find the percentage increase:

\[ \text{Percentage Increase} = \left(\frac{\Delta V}{V}\right) \times 100\% = \left(\frac{1.25V}{V}\right) \times 100\% = 125\% \]

Thus, the volume increases by 125%.