What is the new volume of a right square pyramid with a volume 180 ^3 in. if the lengths of the sides of the base remain fixed but its height is multiplied by a factor of 10?

A. 300 in.
B. 600 in.
C. 1800 in.
D. 6000 in.

1 answer

To solve the problem of finding the new volume of a right square pyramid when its height is multiplied by a factor of 10, we need to use the formula for the volume of a pyramid. The volume \( V \) of a right square pyramid is given by:

\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

Let's break down the information given:

- The original volume of the right square pyramid is \(180 \, \text{in}^3\).
- The height is multiplied by a factor of 10 while the base area remains unchanged.

Let’s assume the original height is \( h \) and the base area is \( A \). The original volume can be written as:

\[ 180 = \frac{1}{3} \times A \times h \]

When the height is multiplied by a factor of 10, the new height becomes \( 10h \). The new volume \( V_{\text{new}} \) will then be:

\[ V_{\text{new}} = \frac{1}{3} \times A \times 10h \]

We can factor out the constants to see that:

\[ V_{\text{new}} = 10 \times \left( \frac{1}{3} \times A \times h \right) \]

Since \( \frac{1}{3} \times A \times h \) is the original volume, which is 180 in³, we get:

\[ V_{\text{new}} = 10 \times 180 \]

\[ V_{\text{new}} = 1800 \, \text{in}^3 \]

Therefore, the new volume of the right square pyramid is:

\[ C. 1800 \, \text{in}^3 \]