Question
se 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)
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)
Answers
GPT-4o mini
To find the surface area of the composite figure formed by the right triangular prism on top of the rectangular prism, we'll need to calculate the surface areas of each individual part and then combine them while taking into account any overlapping or shared areas.
### Step 1: Surface Area of the Rectangular Prism
The surface area \( A \) of a rectangular prism is given by:
\[
A = 2lw + 2lh + 2wh
\]
Where:
- \( l \) is the length,
- \( w \) is the width, and
- \( h \) is the height.
For the rectangular prism:
- Length \( l = 12 \) feet
- Width \( w = 8 \) feet
- Height \( h = 8 \) feet
Calculating the surface area:
\[
A_{\text{rectangular prism}} = 2(12)(8) + 2(12)(8) + 2(8)(8)
\]
\[
= 2(96) + 2(96) + 2(64)
\]
\[
= 192 + 192 + 128
\]
\[
= 512 \text{ square feet}
\]
### Step 2: Surface Area of the Right Triangular Prism
To calculate the surface area of the right triangular prism, we will consider its base and three rectangular faces.
1. **Area of the triangular base**:
The right triangular prism has a height of 15 feet (the hypotenuse) and a width of 8 feet. To find the height of the triangular base, we can use the Pythagorean theorem.
Let \( b \) be the base and \( h_t \) be the height of the triangle:
\[
b^2 + h_t^2 = 15^2
\]
Given the rectangular width (also one dimension of the triangle), \( b = 8 \),
\[
8^2 + h_t^2 = 15^2
\]
\[
64 + h_t^2 = 225
\]
\[
h_t^2 = 161
\]
\[
h_t \approx 12.6886 \text{ feet}
\]
Now, area \( A_{\text{triangle}} \):
\[
A_{\text{triangle}} = \frac{1}{2} \times b \times h_t = \frac{1}{2} \times 8 \times 12.6886 \approx 50.7504 \text{ square feet}
\]
2. **Area of the rectangular faces**:
Each triangular prism has three rectangular faces: the base face covering the triangle, and the two sides.
Area dimensions:
- Front face (triangle base → 8 feet tall while height of the prism = 12 feet)
- Side face → length = height of triangle = 15 feet = 8 + 12.6886 feet each face
So, area contributions:
- Area for both rectangular sides:
- \( 2 \times (8 \times 15) = 240 \text{ square feet} \)
- Area base = \( 50.7504 \)
Therefore, surface area for the triangular prism is \( 240 + 50.7504 = 290.7504 \)
### Final Calculation and Combination of Surface Areas
Total surface area:
\[
\text{Total Surface Area} = A_{\text{rectangular prism}} + A_{\text{triangular prism}} - A_{\text{overlap}}
\]
Where \( A_{\text{overlap}} \) is the area of contact between the two prisms (not visible on the surface).
Since the triangular prism perfectly sits on top of the rectangular dimensions at the top square feet to 8 feet, overlap height remains without contributing another surface area:
\[
= 512 + 290.7504
\]
Thus resulting in \( \text{Total Surface Area} = 802.7504 \)
So, the surface area of the figure is approximately:
\[
\text{Total Surface Area} \approx 803 \text{ square feet}
\]
Hence the answer is **803 square feet**.
### Step 1: Surface Area of the Rectangular Prism
The surface area \( A \) of a rectangular prism is given by:
\[
A = 2lw + 2lh + 2wh
\]
Where:
- \( l \) is the length,
- \( w \) is the width, and
- \( h \) is the height.
For the rectangular prism:
- Length \( l = 12 \) feet
- Width \( w = 8 \) feet
- Height \( h = 8 \) feet
Calculating the surface area:
\[
A_{\text{rectangular prism}} = 2(12)(8) + 2(12)(8) + 2(8)(8)
\]
\[
= 2(96) + 2(96) + 2(64)
\]
\[
= 192 + 192 + 128
\]
\[
= 512 \text{ square feet}
\]
### Step 2: Surface Area of the Right Triangular Prism
To calculate the surface area of the right triangular prism, we will consider its base and three rectangular faces.
1. **Area of the triangular base**:
The right triangular prism has a height of 15 feet (the hypotenuse) and a width of 8 feet. To find the height of the triangular base, we can use the Pythagorean theorem.
Let \( b \) be the base and \( h_t \) be the height of the triangle:
\[
b^2 + h_t^2 = 15^2
\]
Given the rectangular width (also one dimension of the triangle), \( b = 8 \),
\[
8^2 + h_t^2 = 15^2
\]
\[
64 + h_t^2 = 225
\]
\[
h_t^2 = 161
\]
\[
h_t \approx 12.6886 \text{ feet}
\]
Now, area \( A_{\text{triangle}} \):
\[
A_{\text{triangle}} = \frac{1}{2} \times b \times h_t = \frac{1}{2} \times 8 \times 12.6886 \approx 50.7504 \text{ square feet}
\]
2. **Area of the rectangular faces**:
Each triangular prism has three rectangular faces: the base face covering the triangle, and the two sides.
Area dimensions:
- Front face (triangle base → 8 feet tall while height of the prism = 12 feet)
- Side face → length = height of triangle = 15 feet = 8 + 12.6886 feet each face
So, area contributions:
- Area for both rectangular sides:
- \( 2 \times (8 \times 15) = 240 \text{ square feet} \)
- Area base = \( 50.7504 \)
Therefore, surface area for the triangular prism is \( 240 + 50.7504 = 290.7504 \)
### Final Calculation and Combination of Surface Areas
Total surface area:
\[
\text{Total Surface Area} = A_{\text{rectangular prism}} + A_{\text{triangular prism}} - A_{\text{overlap}}
\]
Where \( A_{\text{overlap}} \) is the area of contact between the two prisms (not visible on the surface).
Since the triangular prism perfectly sits on top of the rectangular dimensions at the top square feet to 8 feet, overlap height remains without contributing another surface area:
\[
= 512 + 290.7504
\]
Thus resulting in \( \text{Total Surface Area} = 802.7504 \)
So, the surface area of the figure is approximately:
\[
\text{Total Surface Area} \approx 803 \text{ square feet}
\]
Hence the answer is **803 square feet**.