Question
Find the surface area for a rectangular pyramid. This is a composite 3D figure.
Length is 10 cm, width is 6 cm and height on right side of pyramid shows slant height 13 cm and makes right angle at the bottom. From shows slant height 12.4 cm.
Answer is 262
How do I do this problem
Length is 10 cm, width is 6 cm and height on right side of pyramid shows slant height 13 cm and makes right angle at the bottom. From shows slant height 12.4 cm.
Answer is 262
How do I do this problem
Answers
GPT-4o mini
To find the surface area of a rectangular pyramid, we need to calculate the area of the base and the areas of the triangular faces.
### Step 1: Calculate the area of the base
The base is a rectangle with the given dimensions:
- Length (\( l \)) = 10 cm
- Width (\( w \)) = 6 cm
The area of the base (\( A_{base} \)) is given by:
\[
A_{base} = l \times w = 10 \, \text{cm} \times 6 \, \text{cm} = 60 \, \text{cm}^2
\]
### Step 2: Calculate the area of the triangular faces
A rectangular pyramid has four triangular faces. The slant heights provided are used to calculate the height of these triangles.
- The slant height from the front face is \( 12.4 \) cm.
- The slant height from the side face is \( 13 \) cm.
**2.1: Calculate the area of the front triangular face**
The base of this triangle is the width of the base (\( 6 \) cm) and the height is the slant height \( 12.4 \) cm.
The area of one triangular face is given by:
\[
A_{triangle\_front} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \, \text{cm} \times 12.4 \, \text{cm}
\]
\[
A_{triangle\_front} = \frac{1}{2} \times 6 \times 12.4 = 37.2 \, \text{cm}^2
\]
**2.2: Calculate the area of the back triangular face**
The base of this triangle is again the width of the base (\( 6 \) cm) and the height is the same slant height for the back face, which is also \( 12.4 \, \text{cm} \):
\[
A_{triangle\_back} = A_{triangle\_front} = 37.2 \, \text{cm}^2
\]
**2.3: Calculate the area of the left triangular face**
The base of this triangle is the length of the base (\( 10 \) cm) and the height is the slant height \( 13 \) cm.
\[
A_{triangle\_left} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{cm} \times 13 \, \text{cm}
\]
\[
A_{triangle\_left} = \frac{1}{2} \times 10 \times 13 = 65 \, \text{cm}^2
\]
**2.4: Calculate the area of the right triangular face**
The base of this triangle is again the length of the base (\( 10 \) cm) and the height is also \( 13 \, \text{cm} \).
\[
A_{triangle\_right} = A_{triangle\_left} = 65 \, \text{cm}^2
\]
### Step 3: Total area of triangular faces
Now we will add up the areas of the four triangular faces:
\[
A_{triangles} = A_{triangle\_front} + A_{triangle\_back} + A_{triangle\_left} + A_{triangle\_right}
\]
\[
A_{triangles} = 37.2 + 37.2 + 65 + 65 = 204.4 \, \text{cm}^2
\]
### Step 4: Total Surface Area Calculation
Finally, we add the area of the base and the total area of the triangular faces to get the total surface area of the pyramid.
\[
A_{total} = A_{base} + A_{triangles} = 60 \, \text{cm}^2 + 204.4 \, \text{cm}^2
\]
\[
A_{total} = 264.4 \, \text{cm}^2
\]
### Conclusion:
The computed total surface area of the rectangular pyramid is \( 264.4 \, \text{cm}^2 \).
It appears there is a discrepancy between computed value (264.4 cm²) and the answer you've mentioned (262 cm²). Please verify the slant heights or if there are any rounding instructions you might need to consider.
### Step 1: Calculate the area of the base
The base is a rectangle with the given dimensions:
- Length (\( l \)) = 10 cm
- Width (\( w \)) = 6 cm
The area of the base (\( A_{base} \)) is given by:
\[
A_{base} = l \times w = 10 \, \text{cm} \times 6 \, \text{cm} = 60 \, \text{cm}^2
\]
### Step 2: Calculate the area of the triangular faces
A rectangular pyramid has four triangular faces. The slant heights provided are used to calculate the height of these triangles.
- The slant height from the front face is \( 12.4 \) cm.
- The slant height from the side face is \( 13 \) cm.
**2.1: Calculate the area of the front triangular face**
The base of this triangle is the width of the base (\( 6 \) cm) and the height is the slant height \( 12.4 \) cm.
The area of one triangular face is given by:
\[
A_{triangle\_front} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \, \text{cm} \times 12.4 \, \text{cm}
\]
\[
A_{triangle\_front} = \frac{1}{2} \times 6 \times 12.4 = 37.2 \, \text{cm}^2
\]
**2.2: Calculate the area of the back triangular face**
The base of this triangle is again the width of the base (\( 6 \) cm) and the height is the same slant height for the back face, which is also \( 12.4 \, \text{cm} \):
\[
A_{triangle\_back} = A_{triangle\_front} = 37.2 \, \text{cm}^2
\]
**2.3: Calculate the area of the left triangular face**
The base of this triangle is the length of the base (\( 10 \) cm) and the height is the slant height \( 13 \) cm.
\[
A_{triangle\_left} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{cm} \times 13 \, \text{cm}
\]
\[
A_{triangle\_left} = \frac{1}{2} \times 10 \times 13 = 65 \, \text{cm}^2
\]
**2.4: Calculate the area of the right triangular face**
The base of this triangle is again the length of the base (\( 10 \) cm) and the height is also \( 13 \, \text{cm} \).
\[
A_{triangle\_right} = A_{triangle\_left} = 65 \, \text{cm}^2
\]
### Step 3: Total area of triangular faces
Now we will add up the areas of the four triangular faces:
\[
A_{triangles} = A_{triangle\_front} + A_{triangle\_back} + A_{triangle\_left} + A_{triangle\_right}
\]
\[
A_{triangles} = 37.2 + 37.2 + 65 + 65 = 204.4 \, \text{cm}^2
\]
### Step 4: Total Surface Area Calculation
Finally, we add the area of the base and the total area of the triangular faces to get the total surface area of the pyramid.
\[
A_{total} = A_{base} + A_{triangles} = 60 \, \text{cm}^2 + 204.4 \, \text{cm}^2
\]
\[
A_{total} = 264.4 \, \text{cm}^2
\]
### Conclusion:
The computed total surface area of the rectangular pyramid is \( 264.4 \, \text{cm}^2 \).
It appears there is a discrepancy between computed value (264.4 cm²) and the answer you've mentioned (262 cm²). Please verify the slant heights or if there are any rounding instructions you might need to consider.