Question
ess is building a skateboarding ramp in the shape of a triangular prism with the dimensions shown. If Jess wants to cover all faces of the ramp with plywood, how much plywood will he need?
A triangular prism. The rectangular sides are 12 feet by 3 feet, 12 feet by 5 feet, and 12 feet by 4 feet. The 2 triangular sides have a base of 3 feet and height of 4 feet.
[Not drawn to scale]
96 square feet
144 square feet
156 square feet
168 square feet
A triangular prism. The rectangular sides are 12 feet by 3 feet, 12 feet by 5 feet, and 12 feet by 4 feet. The 2 triangular sides have a base of 3 feet and height of 4 feet.
[Not drawn to scale]
96 square feet
144 square feet
156 square feet
168 square feet
Answers
GPT-4o mini
To find out how much plywood Jess will need to cover all faces of the triangular prism, we need to calculate the surface area of the prism, which consists of the areas of the two triangular bases and the three rectangular sides.
1. **Calculate the area of the triangular base:**
The area \(A\) of a triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the triangular base, we have a base of 3 feet and a height of 4 feet:
\[
A = \frac{1}{2} \times 3 \times 4 = \frac{12}{2} = 6 \text{ square feet}
\]
Since the prism has two triangular faces, the total area of the triangular bases is:
\[
2 \times 6 = 12 \text{ square feet}
\]
2. **Calculate the area of the rectangular sides:**
We have three rectangular sides with the following dimensions:
- \(12 \text{ feet} \times 3 \text{ feet}\)
- \(12 \text{ feet} \times 5 \text{ feet}\)
- \(12 \text{ feet} \times 4 \text{ feet}\)
We can calculate the areas of each rectangle:
- Area of the first rectangle:
\[
12 \times 3 = 36 \text{ square feet}
\]
- Area of the second rectangle:
\[
12 \times 5 = 60 \text{ square feet}
\]
- Area of the third rectangle:
\[
12 \times 4 = 48 \text{ square feet}
\]
Now we sum the areas of the rectangular sides:
\[
36 + 60 + 48 = 144 \text{ square feet}
\]
3. **Total surface area:**
Now, we combine the areas from the triangular bases and the rectangular sides to find the total surface area:
\[
\text{Total area} = \text{Area of triangular bases} + \text{Area of rectangular sides} = 12 + 144 = 156 \text{ square feet}
\]
So, Jess will need **156 square feet** of plywood to cover all faces of the ramp.
The answer is **156 square feet**.
1. **Calculate the area of the triangular base:**
The area \(A\) of a triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the triangular base, we have a base of 3 feet and a height of 4 feet:
\[
A = \frac{1}{2} \times 3 \times 4 = \frac{12}{2} = 6 \text{ square feet}
\]
Since the prism has two triangular faces, the total area of the triangular bases is:
\[
2 \times 6 = 12 \text{ square feet}
\]
2. **Calculate the area of the rectangular sides:**
We have three rectangular sides with the following dimensions:
- \(12 \text{ feet} \times 3 \text{ feet}\)
- \(12 \text{ feet} \times 5 \text{ feet}\)
- \(12 \text{ feet} \times 4 \text{ feet}\)
We can calculate the areas of each rectangle:
- Area of the first rectangle:
\[
12 \times 3 = 36 \text{ square feet}
\]
- Area of the second rectangle:
\[
12 \times 5 = 60 \text{ square feet}
\]
- Area of the third rectangle:
\[
12 \times 4 = 48 \text{ square feet}
\]
Now we sum the areas of the rectangular sides:
\[
36 + 60 + 48 = 144 \text{ square feet}
\]
3. **Total surface area:**
Now, we combine the areas from the triangular bases and the rectangular sides to find the total surface area:
\[
\text{Total area} = \text{Area of triangular bases} + \text{Area of rectangular sides} = 12 + 144 = 156 \text{ square feet}
\]
So, Jess will need **156 square feet** of plywood to cover all faces of the ramp.
The answer is **156 square feet**.