Cylinder Volume Calculation

The volume $V$ of a right circular cylinder is given by the formula:

$V = \pi r^2 h$

where $r$ is the radius and $h$ is the height.

Initial Volume

Let the original radius be $r$ and the original height be $h$. The original volume, $V_1$, is:

$V_1 = \pi r^2 h$

New Volume Calculation

The problem states that the height and radius are doubled. Therefore:

• New radius, $r' = 2r$
• New height, $h' = 2h$

The new volume, $V_2$, is calculated using the new dimensions:

$V_2 = \pi (r')^2 h'$

Substituting the new values:

$V_2 = \pi (2r)^2 (2h) = \pi (4r^2)(2h) = 8 \pi r^2 h$

Percentage Increase Determination

The increase in volume is the difference between the new volume and the original volume:

$\text{Increase} = V_2 - V_1 = 8 \pi r^2 h - \pi r^2 h = 7 \pi r^2 h$

The percentage increase is calculated as:

$\text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Original Volume}} \right) \times 100\%$

Plugging in the values:

$\text{Percentage Increase} = \left( \frac{7 \pi r^2 h}{V_1} \right) \times 100\% = \left( \frac{7 \pi r^2 h}{\pi r^2 h} \right) \times 100\%$

$\text{Percentage Increase} = 7 \times 100\% = 700\%$

Thus, the percentage increase in the cylinder's volume is 700%.