Question

Question
Use the image to answer the question.

An illustration shows a three dimensional composite figure formed by a right triangular prism placed on top of a rectangular prism. The top, right, and front faces are visible. The faces and edges that are not visible are indicated by dashed lines. The rectangular prism has a length of 12 feet and a height of 8 feet. The width of the rectangular prism is 8 feet. The edges of the hidden side face of the right triangular prism align perfectly with the edges of the hidden top face of the rectangular prism. The right triangular prism is aligned to the left of the rectangular prism. The height of the right triangular prism is not shown. The total height of the left side of the figure is 20 feet. The right triangular prism has a width of 8 feet. The hypotenuse side of the right triangular prism has a length of 15 feet.

What is the surface area of the figure?

(1 point)
ft.2

Answer 1

To find the surface area of the composite figure, we need to calculate the surface areas of both the rectangular prism and the right triangular prism, and then combine them while accounting for overlapping areas.

### Step 1: Surface Area of the Rectangular Prism
The formula for the surface area \( A \) of a rectangular prism is given by:

\[
A = 2lw + 2lh + 2wh
\]

where:
- \( l \) is the length
- \( w \) is the width
- \( h \) is the height

From the problem:
- \( l = 12 \, \text{ft} \)
- \( w = 8 \, \text{ft} \)
- \( h = 8 \, \text{ft} \)

Plugging in the values:

\[
A = 2(12)(8) + 2(12)(8) + 2(8)(8)
\]
\[
A = 2(96) + 2(96) + 2(64)
\]
\[
A = 192 + 192 + 128
\]
\[
A = 512 \, \text{ft}^2
\]

### Step 2: Surface Area of the Right Triangular Prism
The surface area \( A \) of a right triangular prism can be calculated using the formula:

\[
A = 2B + Ph
\]

Where:
- \( B \) is the area of the base triangle
- \( P \) is the perimeter of the base triangle
- \( h \) is the height of the prism

#### Finding the Area of the Base Triangle
The triangle is a right triangle where:
- One leg (base) = 8 ft
- We need to find the height of the triangle. Since the hypotenuse is given as 15 ft, we can use the Pythagorean theorem:

Let the unknown height be \( x \):
\[
x^2 + 8^2 = 15^2
\]
\[
x^2 + 64 = 225
\]
\[
x^2 = 161
\]
\[
x = \sqrt{161} \approx 12.688 \, \text{ft}
\]

Now we can calculate the area of the base triangle \( B \):
\[
B = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times \sqrt{161}
\]
\[
B \approx \frac{1}{2} \times 8 \times 12.688 \approx 50.752 \, \text{ft}^2
\]

#### Finding the Perimeter of the Triangle
The perimeter \( P \) of the triangular base is the sum of all sides:
\[
P = 8 + \sqrt{161} + 15
\]
\[
P \approx 8 + 12.688 + 15 \approx 35.688 \, \text{ft}
\]

#### Finding the Height of the Right Triangular Prism
Given that the total height of the left side of the figure is 20 ft and the rectangular prism has a height of 8 ft, the height of the triangular prism is:
\[
h = 20 - 8 = 12 \, \text{ft}
\]

#### Calculate Surface Area of the Right Triangular Prism
Now we can find the surface area of the right triangular prism:

\[
A = 2B + Ph
\]
\[
A \approx 2(50.752) + (35.688)(12)
\]
\[
A \approx 101.504 + 428.256 \approx 529.76 \, \text{ft}^2
\]

### Step 3: Combine Areas
The two prisms are combined, but the top face of the rectangular prism is the same area as the bottom face of the triangular prism (no need to count twice):

\[
\text{Total Area} = \text{Rectangular Prism Area} + \text{Triangular Prism Area} - \text{Area of top face}
\]

The area of the overlapping face (which is the same as the base triangle of the triangular prism):

Area of top face of rectangular prism = Base Area of the triangle:

\[
= 50.752 \, \text{ft}^2
\]

Thus, the combined surface area:

\[
\text{Total Surface Area} = 512 + 529.76 - 50.752
\]
\[
\text{Total Surface Area} = 990.008 \, \text{ft}^2
\]

### Final Answer
The surface area of the figure is approximately **990.01 ft²**.