Question

1. find the volume of the regular pyramid.

(image of a regular pyramid with slant height 13 ft and base edge 7 ft)
options: 91.0 ft³, 318.5 ft³, 212.3 ft³, 637.0 ft³

Explanation:

Step1: Recall the volume formula for a regular pyramid.

The volume \( V \) of a regular pyramid is given by the formula \( V=\frac{1}{3}Bh \), where \( B \) is the area of the base and \( h \) is the height of the pyramid. For a square base (since it's a regular pyramid, the base is a square), if the side length of the base is \( s \), then the area of the base \( B = s^2 \). Wait, but in the diagram, we have a slant edge? Wait, no, the dashed line is the height? Wait, no, the 13 ft: wait, maybe the base is a square with side 7 ft, and the height of the pyramid? Wait, no, maybe I misread. Wait, the regular pyramid: if the base is a square with side length \( s = 7 \) ft, and the height of the pyramid \( h \)? Wait, no, maybe the 13 ft is the slant height? No, no, the formula is \( V=\frac{1}{3}Bh \), where \( B \) is base area, \( h \) is height. Wait, maybe the base is a square with side 7 ft, so \( B = 7\times7 = 49 \) square feet. Then what is the height? Wait, maybe the 13 ft is the height? Wait, no, let's check the options. Let's compute \( \frac{1}{3}\times49\times13 \). Let's calculate that: \( 49\times13 = 637 \), then \( \frac{637}{3}\approx212.3 \). Wait, that matches one of the options. Wait, let's confirm.

Step2: Calculate the base area.

The base is a square with side length \( s = 7 \) ft. So the area of the base \( B = s^2 = 7^2 = 49 \) \( \text{ft}^2 \).

Step3: Apply the volume formula.

Using the volume formula for a pyramid \( V=\frac{1}{3}Bh \), where \( B = 49 \) \( \text{ft}^2 \) and \( h = 13 \) ft (assuming the 13 ft is the height of the pyramid). Then \( V=\frac{1}{3}\times49\times13 \). Calculate \( 49\times13 = 637 \), then \( \frac{637}{3}\approx212.3 \) \( \text{ft}^3 \).

Answer:

212.3 \( \text{ft}^3 \) (corresponding to the option "212.3 \( \text{ft}^3 \)")