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?(1 point)
Answers
GPT-4o mini
To calculate the surface area of the rectangular pyramid, we need to find the areas of the base and the four triangular faces (two on each side).
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 four triangular faces, two triangles corresponding to each pair of opposite sides.
- For the triangle on the longer side (10 ft), with slant height 7 ft:
\[
\text{Area}_{\text{triangle 1}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{ft} \times 7 \, \text{ft} = 35 \, \text{ft}^2
\]
- For the triangle on the shorter side (7 ft), with slant height 8 ft:
\[
\text{Area}_{\text{triangle 2}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \, \text{ft} \times 8 \, \text{ft} = 28 \, \text{ft}^2
\]
3. **Total area of the triangular faces**:
Since there are two triangles for each side, we calculate:
\[
\text{Total Area}_{\text{triangular faces}} = 2 \times \text{Area}_{\text{triangle 1}} + 2 \times \text{Area}_{\text{triangle 2}}
\]
\[
= 2 \times 35 \, \text{ft}^2 + 2 \times 28 \, \text{ft}^2 = 70 \, \text{ft}^2 + 56 \, \text{ft}^2 = 126 \, \text{ft}^2
\]
4. **Total surface area of the pyramid**:
Now, we can add the area of the base to the total area of the triangular faces:
\[
\text{Surface Area} = \text{Area}_{\text{base}} + \text{Total Area}_{\text{triangular faces}}
\]
\[
= 70 \, \text{ft}^2 + 126 \, \text{ft}^2 = 196 \, \text{ft}^2
\]
Thus, the surface area of the rectangular pyramid is \( \boxed{196 \text{ ft}^2} \).
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 four triangular faces, two triangles corresponding to each pair of opposite sides.
- For the triangle on the longer side (10 ft), with slant height 7 ft:
\[
\text{Area}_{\text{triangle 1}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{ft} \times 7 \, \text{ft} = 35 \, \text{ft}^2
\]
- For the triangle on the shorter side (7 ft), with slant height 8 ft:
\[
\text{Area}_{\text{triangle 2}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \, \text{ft} \times 8 \, \text{ft} = 28 \, \text{ft}^2
\]
3. **Total area of the triangular faces**:
Since there are two triangles for each side, we calculate:
\[
\text{Total Area}_{\text{triangular faces}} = 2 \times \text{Area}_{\text{triangle 1}} + 2 \times \text{Area}_{\text{triangle 2}}
\]
\[
= 2 \times 35 \, \text{ft}^2 + 2 \times 28 \, \text{ft}^2 = 70 \, \text{ft}^2 + 56 \, \text{ft}^2 = 126 \, \text{ft}^2
\]
4. **Total surface area of the pyramid**:
Now, we can add the area of the base to the total area of the triangular faces:
\[
\text{Surface Area} = \text{Area}_{\text{base}} + \text{Total Area}_{\text{triangular faces}}
\]
\[
= 70 \, \text{ft}^2 + 126 \, \text{ft}^2 = 196 \, \text{ft}^2
\]
Thus, the surface area of the rectangular pyramid is \( \boxed{196 \text{ ft}^2} \).