Question

what is the volume of the pyramid?
a 864 cm³
b 432 cm³
c 288 cm³
d 108 cm³
(image of a pyramid with labeled dimensions: 12 cm, 9 cm, 8 cm)

Explanation:

Step1: Recall the volume formula for a pyramid

The volume \( V \) of a 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.

Step2: Determine the base area

The base of the pyramid is a rectangle with length \( l = 12\space\text{cm} \) and width \( w = 9\space\text{cm} \). The area of a rectangle is \( B=l\times w \), so \( B = 12\times9=108\space\text{cm}^2 \).

Step3: Identify the height of the pyramid

From the diagram, the height \( h \) of the pyramid is \( 8\space\text{cm} \).

Step4: Calculate the volume

Substitute \( B = 108\space\text{cm}^2 \) and \( h = 8\space\text{cm} \) into the volume formula:

$$ V=\frac{1}{3}\times108\times8 $$

First, calculate \( \frac{1}{3}\times108 = 36 \), then multiply by \( 8 \): \( 36\times8 = 288\space\text{cm}^3 \). Wait, no, wait, maybe I misread the height? Wait, no, wait the diagram: Wait, maybe the base is a different shape? Wait, no, the base is a rectangle with length 12, width 9, and the height of the pyramid is 8? Wait, no, maybe I made a mistake. Wait, let's check again. Wait, the formula is \( V=\frac{1}{3}Bh \). Wait, base area: 129=108. Height: 8. Then \( \frac{1}{3}*108*8 = 36*8 = 288 \). But wait, the options have 288 as option C. Wait, but maybe the height is different? Wait, maybe the slant height? No, the height is the perpendicular height. Wait, maybe I misread the diagram. Wait, the diagram shows 12 cm, 9 cm, and 8 cm. Wait, maybe the base is a rectangle with length 12, width 9, and height 8. Then volume is \( \frac{1}{3}*12*9*8 \). Let's compute that: 129=108, 1088=864, 864/3=288? Wait, no, 129*8=864, divided by 3 is 288. Wait, but the options: A is 864, B is 432, C is 288, D is 108. So the correct calculation is \( V=\frac{1}{3}\times12\times9\times8=\frac{1}{3}\times864 = 288\space\text{cm}^3 \), which is option C. Wait, but the original selected option was A, which is wrong. So let's redo:

Wait, maybe the height is 8, base length 12, base width 9. Then:

\( V=\frac{1}{3} \times (\text{base area}) \times \text{height} \)

Base area = 12 * 9 = 108

Height = 8

So \( V = \frac{1}{3} \times 108 \times 8 = 36 \times 8 = 288 \). So the correct answer is C. 288 cm³.

Answer:

C. 288 cm³