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. what is the surface area of the pyramid?
Answers
Answered by
GPT-4o mini
To calculate the surface area of the rectangular pyramid, we need to find the area of the base and the area of the triangular faces.
1. **Base Area**:
The base of the pyramid is rectangular with dimensions \(10 \, \text{ft} \times 7 \, \text{ft}\).
\[
\text{Base Area} = \text{Length} \times \text{Width} = 10 \, \text{ft} \times 7 \, \text{ft} = 70 \, \text{ft}^2
\]
2. **Triangular Faces**:
There are four triangular faces in the pyramid:
- Two faces have a base of \(10 \, \text{ft}\) (the longer side) and a slant height of \(7 \, \text{ft}\).
- Two faces have a base of \(7 \, \text{ft}\) (the shorter side) and a slant height of \(8 \, \text{ft}\).
- **Area of triangular faces with base 10 ft**:
\[
\text{Area}_{10 \, \text{ft}} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 10 \, \text{ft} \times 7 \, \text{ft} = 35 \, \text{ft}^2
\]
Since there are two of these triangular faces:
\[
\text{Total Area}_{10 \, \text{ft}} = 2 \times 35 \, \text{ft}^2 = 70 \, \text{ft}^2
\]
- **Area of triangular faces with base 7 ft**:
\[
\text{Area}_{7 \, \text{ft}} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 7 \, \text{ft} \times 8 \, \text{ft} = 28 \, \text{ft}^2
\]
Since there are two of these triangular faces:
\[
\text{Total Area}_{7 \, \text{ft}} = 2 \times 28 \, \text{ft}^2 = 56 \, \text{ft}^2
\]
3. **Total Surface Area**:
The total surface area of the pyramid is the sum of the base area and the areas of the triangular faces.
\[
\text{Total Surface Area} = \text{Base Area} + \text{Total Area}_{10 \, \text{ft}} + \text{Total Area}_{7 \, \text{ft}}
\]
\[
\text{Total Surface Area} = 70 \, \text{ft}^2 + 70 \, \text{ft}^2 + 56 \, \text{ft}^2 = 196 \, \text{ft}^2
\]
Thus, the surface area of the pyramid is \(\boxed{196 \, \text{ft}^2}\).
1. **Base Area**:
The base of the pyramid is rectangular with dimensions \(10 \, \text{ft} \times 7 \, \text{ft}\).
\[
\text{Base Area} = \text{Length} \times \text{Width} = 10 \, \text{ft} \times 7 \, \text{ft} = 70 \, \text{ft}^2
\]
2. **Triangular Faces**:
There are four triangular faces in the pyramid:
- Two faces have a base of \(10 \, \text{ft}\) (the longer side) and a slant height of \(7 \, \text{ft}\).
- Two faces have a base of \(7 \, \text{ft}\) (the shorter side) and a slant height of \(8 \, \text{ft}\).
- **Area of triangular faces with base 10 ft**:
\[
\text{Area}_{10 \, \text{ft}} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 10 \, \text{ft} \times 7 \, \text{ft} = 35 \, \text{ft}^2
\]
Since there are two of these triangular faces:
\[
\text{Total Area}_{10 \, \text{ft}} = 2 \times 35 \, \text{ft}^2 = 70 \, \text{ft}^2
\]
- **Area of triangular faces with base 7 ft**:
\[
\text{Area}_{7 \, \text{ft}} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 7 \, \text{ft} \times 8 \, \text{ft} = 28 \, \text{ft}^2
\]
Since there are two of these triangular faces:
\[
\text{Total Area}_{7 \, \text{ft}} = 2 \times 28 \, \text{ft}^2 = 56 \, \text{ft}^2
\]
3. **Total Surface Area**:
The total surface area of the pyramid is the sum of the base area and the areas of the triangular faces.
\[
\text{Total Surface Area} = \text{Base Area} + \text{Total Area}_{10 \, \text{ft}} + \text{Total Area}_{7 \, \text{ft}}
\]
\[
\text{Total Surface Area} = 70 \, \text{ft}^2 + 70 \, \text{ft}^2 + 56 \, \text{ft}^2 = 196 \, \text{ft}^2
\]
Thus, the surface area of the pyramid is \(\boxed{196 \, \text{ft}^2}\).
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