Question
A rectangular pyramid has a base with sides 10 ft. and 7 ft. The slant height where the base has the longer side is 7 ft., and the slant height where the base has the shorter side is 8 ft. Which of the following correctly calculates the surface area of the pyramid?
a. 70 ft.^2
b. 202 ft.^2
c. 196 ft.^2
d. 126 ft.^2
a. 70 ft.^2
b. 202 ft.^2
c. 196 ft.^2
d. 126 ft.^2
Answers
GPT-4o mini
To calculate the surface area of a rectangular pyramid, we need to find both the area of the base and the areas of the four triangular faces.
1. **Area of the base**:
The area of the rectangular base can be calculated using the formula:
\[
\text{Area}_{\text{base}} = \text{length} \times \text{width} = 10 \, \text{ft} \times 7 \, \text{ft} = 70 \, \text{ft}^2
\]
2. **Area of the triangular faces**:
There are two triangles for the longer side (10 ft) and two triangles for the shorter side (7 ft).
- **For the triangles on the longer side (10 ft)**:
The slant height is 7 ft.
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{ft} \times 7 \, \text{ft} = 35 \, \text{ft}^2
\]
Since there are two such triangles:
\[
\text{Total area}_{\text{long side}} = 2 \times 35 \, \text{ft}^2 = 70 \, \text{ft}^2
\]
- **For the triangles on the shorter side (7 ft)**:
The slant height is 8 ft.
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \, \text{ft} \times 8 \, \text{ft} = 28 \, \text{ft}^2
\]
Since there are two such triangles:
\[
\text{Total area}_{\text{short side}} = 2 \times 28 \, \text{ft}^2 = 56 \, \text{ft}^2
\]
3. **Total surface area**:
Now we can add the area of the base and the areas of the triangular faces:
\[
\text{Surface area} = \text{Area}_{\text{base}} + \text{Total area}_{\text{long side}} + \text{Total area}_{\text{short side}}
\]
\[
\text{Surface area} = 70 \, \text{ft}^2 + 70 \, \text{ft}^2 + 56 \, \text{ft}^2 = 196 \, \text{ft}^2
\]
Thus, the surface area of the pyramid is **196 ft²**.
The correct answer is:
**c. 196 ft²**.
1. **Area of the base**:
The area of the rectangular base can be calculated using the formula:
\[
\text{Area}_{\text{base}} = \text{length} \times \text{width} = 10 \, \text{ft} \times 7 \, \text{ft} = 70 \, \text{ft}^2
\]
2. **Area of the triangular faces**:
There are two triangles for the longer side (10 ft) and two triangles for the shorter side (7 ft).
- **For the triangles on the longer side (10 ft)**:
The slant height is 7 ft.
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{ft} \times 7 \, \text{ft} = 35 \, \text{ft}^2
\]
Since there are two such triangles:
\[
\text{Total area}_{\text{long side}} = 2 \times 35 \, \text{ft}^2 = 70 \, \text{ft}^2
\]
- **For the triangles on the shorter side (7 ft)**:
The slant height is 8 ft.
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \, \text{ft} \times 8 \, \text{ft} = 28 \, \text{ft}^2
\]
Since there are two such triangles:
\[
\text{Total area}_{\text{short side}} = 2 \times 28 \, \text{ft}^2 = 56 \, \text{ft}^2
\]
3. **Total surface area**:
Now we can add the area of the base and the areas of the triangular faces:
\[
\text{Surface area} = \text{Area}_{\text{base}} + \text{Total area}_{\text{long side}} + \text{Total area}_{\text{short side}}
\]
\[
\text{Surface area} = 70 \, \text{ft}^2 + 70 \, \text{ft}^2 + 56 \, \text{ft}^2 = 196 \, \text{ft}^2
\]
Thus, the surface area of the pyramid is **196 ft²**.
The correct answer is:
**c. 196 ft²**.