Question
Find the surface area of the rectangle Base pyramid. Base is 1.2m and 0.9m. One slant height is 3.8m and the other slant height is 3.82m.
15.7
1.08
7.98
9.06
15.7
1.08
7.98
9.06
Answers
GPT-4o mini
To find the surface area of the rectangular pyramid with a base and two different slant heights, we will follow these steps:
1. **Calculate the area of the base.**
2. **Calculate the area of the four triangular faces.**
3. **Sum these areas to get the total surface area.**
### Step 1: Area of the Base
The area \( A_{\text{base}} \) of the rectangular base is given by:
\[
A_{\text{base}} = \text{length} \times \text{width} = 1.2 \, \text{m} \times 0.9 \, \text{m} = 1.08 \, \text{m}^2
\]
### Step 2: Area of the Triangular Faces
There are two triangular faces, each with their own slant heights.
- **For the triangle with slant height 3.8 m (base length 1.2 m):**
The area \( A_1 \) of a triangle is given by:
\[
A_1 = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1.2 \, \text{m} \times 3.8 \, \text{m} = 2.28 \, \text{m}^2
\]
- **For the triangle with slant height 3.82 m (base length 0.9 m):**
The area \( A_2 \) of this triangle is:
\[
A_2 = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 0.9 \, \text{m} \times 3.82 \, \text{m} = 1.7175 \, \text{m}^2
\]
### Step 3: Total Area of the Triangular Faces
There are two triangular faces for each pair of base lengths, so we need to multiply each triangular area by 2:
- Total area for the triangle with slant height 3.8 m:
\[
\text{Total area for } A_1 = 2 \times 2.28 = 4.56 \, \text{m}^2
\]
- Total area for the triangle with slant height 3.82 m:
\[
\text{Total area for } A_2 = 2 \times 1.7175 = 3.435 \, \text{m}^2
\]
### Step 4: Total Surface Area
Finally, we add the area of the base to the total area of the triangular faces:
\[
\text{Total Surface Area} = A_{\text{base}} + \text{Total area for } A_1 + \text{Total area for } A_2
\]
\[
\text{Total Surface Area} = 1.08 + 4.56 + 3.435 = 9.075 \, \text{m}^2
\]
### Conclusion
The total surface area of the rectangular pyramid is approximately \( 9.08 \, \text{m}^2 \). Among the provided options, the closest value is **9.06**.
1. **Calculate the area of the base.**
2. **Calculate the area of the four triangular faces.**
3. **Sum these areas to get the total surface area.**
### Step 1: Area of the Base
The area \( A_{\text{base}} \) of the rectangular base is given by:
\[
A_{\text{base}} = \text{length} \times \text{width} = 1.2 \, \text{m} \times 0.9 \, \text{m} = 1.08 \, \text{m}^2
\]
### Step 2: Area of the Triangular Faces
There are two triangular faces, each with their own slant heights.
- **For the triangle with slant height 3.8 m (base length 1.2 m):**
The area \( A_1 \) of a triangle is given by:
\[
A_1 = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1.2 \, \text{m} \times 3.8 \, \text{m} = 2.28 \, \text{m}^2
\]
- **For the triangle with slant height 3.82 m (base length 0.9 m):**
The area \( A_2 \) of this triangle is:
\[
A_2 = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 0.9 \, \text{m} \times 3.82 \, \text{m} = 1.7175 \, \text{m}^2
\]
### Step 3: Total Area of the Triangular Faces
There are two triangular faces for each pair of base lengths, so we need to multiply each triangular area by 2:
- Total area for the triangle with slant height 3.8 m:
\[
\text{Total area for } A_1 = 2 \times 2.28 = 4.56 \, \text{m}^2
\]
- Total area for the triangle with slant height 3.82 m:
\[
\text{Total area for } A_2 = 2 \times 1.7175 = 3.435 \, \text{m}^2
\]
### Step 4: Total Surface Area
Finally, we add the area of the base to the total area of the triangular faces:
\[
\text{Total Surface Area} = A_{\text{base}} + \text{Total area for } A_1 + \text{Total area for } A_2
\]
\[
\text{Total Surface Area} = 1.08 + 4.56 + 3.435 = 9.075 \, \text{m}^2
\]
### Conclusion
The total surface area of the rectangular pyramid is approximately \( 9.08 \, \text{m}^2 \). Among the provided options, the closest value is **9.06**.