Question
Marlon built a ramp to put in front of the curb near his driveway so he could get to the sidewalk more easily from the street on his bike.
A rectangular prism with a length of 6 inches, width of 18 inches, and height of 6 inches. A triangular prism. The triangular sides have a base of 8 inches and height of 6 inches. The prism has a height of 18 inches.
If the ramp includes the flat piece as well as the angled piece and is made entirely out of concrete, what is the total amount of concrete in the ramp?
A rectangular prism with a length of 6 inches, width of 18 inches, and height of 6 inches. A triangular prism. The triangular sides have a base of 8 inches and height of 6 inches. The prism has a height of 18 inches.
If the ramp includes the flat piece as well as the angled piece and is made entirely out of concrete, what is the total amount of concrete in the ramp?
Answers
GPT-4o mini
To find the total amount of concrete used in Marlon's ramp, we need to calculate the volume of both the rectangular prism and the triangular prism and then add those volumes together.
### Step 1: Calculate the volume of the rectangular prism
The formula for the volume \( V \) of a rectangular prism is given by:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
For the rectangular prism:
- Length = 6 inches
- Width = 18 inches
- Height = 6 inches
Substituting in these values:
\[
V = 6 \times 18 \times 6 = 648 \text{ cubic inches}
\]
### Step 2: Calculate the volume of the triangular prism
The formula for the volume \( V \) of a triangular prism is given by:
\[
V = \text{Area of the base} \times \text{height}
\]
First, we need to find the area of the triangular base, which can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the triangular base of the triangular prism:
- Base = 8 inches
- Height = 6 inches
Calculating the area:
\[
\text{Area} = \frac{1}{2} \times 8 \times 6 = 24 \text{ square inches}
\]
Next, we can find the volume of the triangular prism. The height of the prism is 18 inches.
Substituting the area into the volume formula:
\[
V = 24 \times 18 = 432 \text{ cubic inches}
\]
### Step 3: Calculate the total volume of the ramp
Now we can find the total volume of concrete in the ramp by adding the volumes of the rectangular and triangular prisms:
\[
\text{Total Volume} = \text{Volume of Rectangular Prism} + \text{Volume of Triangular Prism}
\]
Calculating the total:
\[
\text{Total Volume} = 648 + 432 = 1080 \text{ cubic inches}
\]
### Conclusion
The total amount of concrete in the ramp is
\[
\boxed{1080} \text{ cubic inches}
\]
### Step 1: Calculate the volume of the rectangular prism
The formula for the volume \( V \) of a rectangular prism is given by:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
For the rectangular prism:
- Length = 6 inches
- Width = 18 inches
- Height = 6 inches
Substituting in these values:
\[
V = 6 \times 18 \times 6 = 648 \text{ cubic inches}
\]
### Step 2: Calculate the volume of the triangular prism
The formula for the volume \( V \) of a triangular prism is given by:
\[
V = \text{Area of the base} \times \text{height}
\]
First, we need to find the area of the triangular base, which can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the triangular base of the triangular prism:
- Base = 8 inches
- Height = 6 inches
Calculating the area:
\[
\text{Area} = \frac{1}{2} \times 8 \times 6 = 24 \text{ square inches}
\]
Next, we can find the volume of the triangular prism. The height of the prism is 18 inches.
Substituting the area into the volume formula:
\[
V = 24 \times 18 = 432 \text{ cubic inches}
\]
### Step 3: Calculate the total volume of the ramp
Now we can find the total volume of concrete in the ramp by adding the volumes of the rectangular and triangular prisms:
\[
\text{Total Volume} = \text{Volume of Rectangular Prism} + \text{Volume of Triangular Prism}
\]
Calculating the total:
\[
\text{Total Volume} = 648 + 432 = 1080 \text{ cubic inches}
\]
### Conclusion
The total amount of concrete in the ramp is
\[
\boxed{1080} \text{ cubic inches}
\]