Question
Esther wants to construct a futon couch by stuffing the cushions and the back with soft foam. How much material does Esther need to create the couch if it has the dimensions shown below?
Do you need to find the surface area or volume of the couch in order to solve this problem?
L-shaped prism. The base has side lengths labeled 20 in, 10 in, 1 unlabeled side length, 20 in, 5 in, 30 in. The height of the prism is 55 in.
Volume
Surface area
Excellent!
This futon couch is in the shape of a prism. To find its volume, you can multiply the area of the base by the height. Start by finding the area of the base on paper, then record your answer on Zearn. You can use the calculator to help you solve.
L-shaped prism. The base has side lengths labeled 20 in, 10 in, 1 unlabeled side length, 20 in, 5 in, 30 in. The height of the prism is 55 in.
Area of the base,
B Height of the prism,
h Volume
V=B·h
in
2
Do you need to find the surface area or volume of the couch in order to solve this problem?
L-shaped prism. The base has side lengths labeled 20 in, 10 in, 1 unlabeled side length, 20 in, 5 in, 30 in. The height of the prism is 55 in.
Volume
Surface area
Excellent!
This futon couch is in the shape of a prism. To find its volume, you can multiply the area of the base by the height. Start by finding the area of the base on paper, then record your answer on Zearn. You can use the calculator to help you solve.
L-shaped prism. The base has side lengths labeled 20 in, 10 in, 1 unlabeled side length, 20 in, 5 in, 30 in. The height of the prism is 55 in.
Area of the base,
B Height of the prism,
h Volume
V=B·h
in
2
Answers
GPT-4o mini
To calculate the volume of the L-shaped prism that Esther wants to construct for her futon couch, you'll need to follow these steps:
1. **Find the Area of the Base:**
The base is L-shaped, and we can break it down into a rectangle and a smaller rectangle (or square) to find the total area.
Let's visualize the base dimensions:
- One part of the L-shape has dimensions 20 in (length) and 10 in (width).
- The smaller part of the L-shape has dimensions 5 in (width) and an unlabeled side which we can assume is aligned with the 20 in side of the first rectangle.
First, calculate the area of the larger rectangle (the part that is 20 in by 10 in):
\[
A_1 = 20 \, \text{in} \times 10 \, \text{in} = 200 \, \text{in}^2
\]
Now calculate the area of the smaller rectangle (5 in by the remaining length):
- Since the total length of the L-shaped base must be 30 in, we can find the other dimension.
- The side opposite the 5 in width will be \( 30 \, \text{in} - 20 \, \text{in} = 10 \, \text{in} \).
Calculate the area of this rectangular part:
\[
A_2 = 5 \, \text{in} \times 10 \, \text{in} = 50 \, \text{in}^2
\]
Now, add both areas together to get the total area of the base \( B \):
\[
B = A_1 + A_2 = 200 \, \text{in}^2 + 50 \, \text{in}^2 = 250 \, \text{in}^2
\]
2. **Height of the Prism:**
The height \( h \) of the prism is given as 55 in.
3. **Calculate the Volume:**
Now, use the formula for volume:
\[
V = B \cdot h
\]
Substitute in the values:
\[
V = 250 \, \text{in}^2 \times 55 \, \text{in} = 13750 \, \text{in}^3
\]
Therefore, the total volume of foam needed to stuff the cushions and back of the futon couch is **13,750 cubic inches** (in³).
To summarize, you calculated the **volume** of the couch, not the surface area, to determine how much foam material is needed.
1. **Find the Area of the Base:**
The base is L-shaped, and we can break it down into a rectangle and a smaller rectangle (or square) to find the total area.
Let's visualize the base dimensions:
- One part of the L-shape has dimensions 20 in (length) and 10 in (width).
- The smaller part of the L-shape has dimensions 5 in (width) and an unlabeled side which we can assume is aligned with the 20 in side of the first rectangle.
First, calculate the area of the larger rectangle (the part that is 20 in by 10 in):
\[
A_1 = 20 \, \text{in} \times 10 \, \text{in} = 200 \, \text{in}^2
\]
Now calculate the area of the smaller rectangle (5 in by the remaining length):
- Since the total length of the L-shaped base must be 30 in, we can find the other dimension.
- The side opposite the 5 in width will be \( 30 \, \text{in} - 20 \, \text{in} = 10 \, \text{in} \).
Calculate the area of this rectangular part:
\[
A_2 = 5 \, \text{in} \times 10 \, \text{in} = 50 \, \text{in}^2
\]
Now, add both areas together to get the total area of the base \( B \):
\[
B = A_1 + A_2 = 200 \, \text{in}^2 + 50 \, \text{in}^2 = 250 \, \text{in}^2
\]
2. **Height of the Prism:**
The height \( h \) of the prism is given as 55 in.
3. **Calculate the Volume:**
Now, use the formula for volume:
\[
V = B \cdot h
\]
Substitute in the values:
\[
V = 250 \, \text{in}^2 \times 55 \, \text{in} = 13750 \, \text{in}^3
\]
Therefore, the total volume of foam needed to stuff the cushions and back of the futon couch is **13,750 cubic inches** (in³).
To summarize, you calculated the **volume** of the couch, not the surface area, to determine how much foam material is needed.