Questions: What If? If the height of the pyramid were increased to 516 ft and the height to base area ratio of the pyramid were kept constant, by what percentage would the volume of the pyramid increase?

Solution Steps

To solve this problem, we need to understand the relationship between the height and the volume of a pyramid. The volume \( V \) of a pyramid is given by the formula \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \). If the height is increased while keeping the height to base area ratio constant, the base area will scale proportionally with the square of the height. We can then calculate the percentage increase in volume.

We need to determine the percentage increase in the volume of a pyramid when its height is increased to 516 ft, while keeping the height to base area ratio constant.

The volume \( V \) of a pyramid is given by: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

Let the initial height of the pyramid be \( h_1 \) and the final height be \( h_2 = 516 \) ft.

Let the height to base area ratio be \( k \). Therefore, we have: \[ \frac{h_1}{A_1} = \frac{h_2}{A_2} = k \] where \( A_1 \) and \( A_2 \) are the base areas corresponding to heights \( h_1 \) and \( h_2 \) respectively.

From the ratio, we get: \[ A_1 = \frac{h_1}{k} \] \[ A_2 = \frac{h_2}{k} \]

The initial volume \( V_1 \) is: \[ V_1 = \frac{1}{3} \times A_1 \times h_1 = \frac{1}{3} \times \frac{h_1}{k} \times h_1 = \frac{h_1^2}{3k} \]

The final volume \( V_2 \) is: \[ V_2 = \frac{1}{3} \times A_2 \times h_2 = \frac{1}{3} \times \frac{h_2}{k} \times h_2 = \frac{h_2^2}{3k} \]

The percentage increase in volume is given by: \[ \text{Percentage Increase} = \left( \frac{V_2 - V_1}{V_1} \right) \times 100\% \]

Substitute \( V_1 \) and \( V_2 \): \[ \text{Percentage Increase} = \left( \frac{\frac{h_2^2}{3k} - \frac{h_1^2}{3k}}{\frac{h_1^2}{3k}} \right) \times 100\% \] \[ \text{Percentage Increase} = \left( \frac{h_2^2 - h_1^2}{h_1^2} \right) \times 100\% \]

Let \( h_1 \) be the initial height and \( h_2 = 516 \) ft. We need the initial height \( h_1 \) to proceed, but since it is not given, we assume it is \( h_1 \).

\[ \text{Percentage Increase} = \left( \frac{516^2 - h_1^2}{h_1^2} \right) \times 100\% \]

Final Answer