Question

find the volume of the composite shapes. show all your work.
12.
13.

Explanation:

Problem 12:

Step 1: Volume of the rectangular prism

The formula for the volume of a rectangular prism is \( V_{prism} = l \times w \times h \). Here, \( l = 4 \, \text{cm} \), \( w = 4 \, \text{cm} \), and \( h = 6 \, \text{cm} \).
\[
V_{prism} = 4 \times 4 \times 6 = 96 \, \text{cm}^3
\]

Step 2: Volume of the square pyramid

The formula for the volume of a pyramid is \( V_{pyramid} = \frac{1}{3} \times B \times h \), where \( B \) is the area of the base. The base is a square with side \( 4 \, \text{cm} \), so \( B = 4 \times 4 = 16 \, \text{cm}^2 \), and the height \( h = 5 \, \text{cm} \).
\[
V_{pyramid} = \frac{1}{3} \times 16 \times 5 = \frac{80}{3} \approx 26.67 \, \text{cm}^3
\]

Step 3: Total volume of the composite shape

Add the volumes of the prism and the pyramid.
\[
V_{total} = 96 + \frac{80}{3} = \frac{288 + 80}{3} = \frac{368}{3} \approx 122.67 \, \text{cm}^3
\]
(Note: The original work had an error in the base area of the pyramid; it should be \( 16 \, \text{cm}^2 \) instead of \( 9 \, \text{cm}^2 \).)

Step 1: Volume of the rectangular prism

The formula for the volume of a rectangular prism is \( V_{prism} = l \times w \times h \). Here, \( l = 14 \, \text{ft} \), \( w = 8 \, \text{ft} \), and \( h = 12 \, \text{ft} \).
\[
V_{prism} = 14 \times 8 \times 12 = 1344 \, \text{ft}^3
\]

Step 2: Volume of the cylinder

The formula for the volume of a cylinder is \( V_{cylinder} = \pi r^2 h \). Here, \( r = 4 \, \text{ft} \) and \( h = 8 \, \text{ft} \).
\[
V_{cylinder} = \pi \times 4^2 \times 8 = \pi \times 16 \times 8 = 128\pi \approx 402.12 \, \text{ft}^3
\]

Step 3: Total volume of the composite shape

Add the volumes of the prism and the cylinder.
\[
V_{total} = 1344 + 128\pi \approx 1344 + 402.12 = 1746.12 \, \text{ft}^3
\]

Answer:

The volume of the composite shape in problem 12 is \(\frac{368}{3} \approx 122.67 \, \text{cm}^3\).

Problem 13: