Question

find the volume of the solid below, composed of two pyramids connected by the same vertical square base. the volume of the solid is ______ cubic centimeters. the solution is

Explanation:

Answer:

To find the volume of the solid composed of two pyramids with the same square base, we use the formula for the volume of a pyramid, \( V = \frac{1}{3}Bh \), where \( B \) is the area of the base and \( h \) is the height.

Step 1: Determine the base area

The base is a square with side length \( 8 \, \text{cm} \).
Area of the base, \( B = 8 \times 8 = 64 \, \text{cm}^2 \).

Step 2: Determine the heights of the two pyramids

From the diagram, the heights of the two pyramids are \( 6 \, \text{cm} \) and \( 12 \, \text{cm} \).

Step 3: Calculate the volume of each pyramid

• Volume of the first pyramid (height \( 6 \, \text{cm} \)):

\( V_1 = \frac{1}{3} \times 64 \times 6 \)
\( V_1 = \frac{1}{3} \times 384 = 128 \, \text{cm}^3 \)

• Volume of the second pyramid (height \( 12 \, \text{cm} \)):

\( V_2 = \frac{1}{3} \times 64 \times 12 \)
\( V_2 = \frac{1}{3} \times 768 = 256 \, \text{cm}^3 \)

Step 4: Total volume of the solid

Add the volumes of the two pyramids:
\( V_{\text{total}} = V_1 + V_2 = 128 + 256 = 384 \, \text{cm}^3 \)

The volume of the solid is \(\boldsymbol{384}\) cubic centimeters.