Question
A rectangular pyramid has a base with sides 10 ft. and 8 ft. The slant height where the base has the longer side is 9 ft., and the slant height where the base has the shorter side is 10 ft. Which of the following correctly calculates the surface area of the pyramid?(1 point)Responses80 ft.280 ft. squared250 ft.2250 ft. squared280 ft.2280 ft. squared170 ft.2
Answers
GPT-4o mini
To find the surface area of the rectangular pyramid, we need to calculate the area of the base and the areas of the four triangular faces.
1. **Base Area**:
The base is a rectangle with dimensions 10 ft and 8 ft.
\[
\text{Base Area} = \text{length} \times \text{width} = 10 \text{ ft} \times 8 \text{ ft} = 80 \text{ ft}^2
\]
2. **Triangular Face Areas**:
The pyramid has two triangular faces corresponding to the long side (10 ft) and two faces for the short side (8 ft).
- **Area of the triangular face with the longer base (10 ft)**:
The slant height for this triangular face is 9 ft.
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \text{ ft} \times 9 \text{ ft} = 45 \text{ ft}^2
\]
Since there are two such triangles:
\[
\text{Total area for longer base} = 2 \times 45 \text{ ft}^2 = 90 \text{ ft}^2
\]
- **Area of the triangular face with the shorter base (8 ft)**:
The slant height for this triangular face is 10 ft.
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \text{ ft} \times 10 \text{ ft} = 40 \text{ ft}^2
\]
Since there are two such triangles:
\[
\text{Total area for shorter base} = 2 \times 40 \text{ ft}^2 = 80 \text{ ft}^2
\]
3. **Total Surface Area**:
Now, we can sum the areas:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Area of both longer side triangles} + \text{Area of both shorter side triangles}
\]
\[
\text{Total Surface Area} = 80 \text{ ft}^2 + 90 \text{ ft}^2 + 80 \text{ ft}^2 = 250 \text{ ft}^2
\]
Therefore, the surface area of the pyramid is **250 ft²**.
The correct response is **250 ft.²**.
1. **Base Area**:
The base is a rectangle with dimensions 10 ft and 8 ft.
\[
\text{Base Area} = \text{length} \times \text{width} = 10 \text{ ft} \times 8 \text{ ft} = 80 \text{ ft}^2
\]
2. **Triangular Face Areas**:
The pyramid has two triangular faces corresponding to the long side (10 ft) and two faces for the short side (8 ft).
- **Area of the triangular face with the longer base (10 ft)**:
The slant height for this triangular face is 9 ft.
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \text{ ft} \times 9 \text{ ft} = 45 \text{ ft}^2
\]
Since there are two such triangles:
\[
\text{Total area for longer base} = 2 \times 45 \text{ ft}^2 = 90 \text{ ft}^2
\]
- **Area of the triangular face with the shorter base (8 ft)**:
The slant height for this triangular face is 10 ft.
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \text{ ft} \times 10 \text{ ft} = 40 \text{ ft}^2
\]
Since there are two such triangles:
\[
\text{Total area for shorter base} = 2 \times 40 \text{ ft}^2 = 80 \text{ ft}^2
\]
3. **Total Surface Area**:
Now, we can sum the areas:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Area of both longer side triangles} + \text{Area of both shorter side triangles}
\]
\[
\text{Total Surface Area} = 80 \text{ ft}^2 + 90 \text{ ft}^2 + 80 \text{ ft}^2 = 250 \text{ ft}^2
\]
Therefore, the surface area of the pyramid is **250 ft²**.
The correct response is **250 ft.²**.