Question
A bluetooth speaker is in the shape of a cube with an edge length of 4.5 inches.
(3 points)
**Do not round the answers.**
Determine the volume of the speaker.
cubic inches
The speaker is sold in a container in the shape of a square prism with a base of 25 square inches and a height of 6.25 inches.
Determine the volume of the container.
cubic inches
How much empty space remains in the container after the speaker is placed in the package?
cubic inches
(3 points)
**Do not round the answers.**
Determine the volume of the speaker.
cubic inches
The speaker is sold in a container in the shape of a square prism with a base of 25 square inches and a height of 6.25 inches.
Determine the volume of the container.
cubic inches
How much empty space remains in the container after the speaker is placed in the package?
cubic inches
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine the volume of the Bluetooth speaker, we start with the formula for the volume of a cube, which is given by:
\[
V = s^3
\]
where \( s \) is the length of an edge of the cube. Given that the edge length is 4.5 inches, we can calculate the volume of the speaker:
\[
V_{\text{speaker}} = (4.5)^3
\]
Calculating \( 4.5^3 \):
\[
4.5^3 = 4.5 \times 4.5 \times 4.5
\]
Calculating \( 4.5 \times 4.5 = 20.25 \):
\[
20.25 \times 4.5 = 91.125
\]
Therefore, the volume of the speaker is:
\[
V_{\text{speaker}} = 91.125 \text{ cubic inches}
\]
Next, we calculate the volume of the container, which is in the shape of a square prism. The formula for the volume of a prism is given by:
\[
V = B \times h
\]
where \( B \) is the area of the base and \( h \) is the height. The base area \( B \) is 25 square inches and the height \( h \) is 6.25 inches. Therefore:
\[
V_{\text{container}} = 25 \times 6.25
\]
Calculating this gives:
\[
25 \times 6.25 = 156.25
\]
Thus, the volume of the container is:
\[
V_{\text{container}} = 156.25 \text{ cubic inches}
\]
Now, to find out how much empty space remains in the container after placing the speaker inside, we subtract the volume of the speaker from the volume of the container:
\[
\text{Empty space} = V_{\text{container}} - V_{\text{speaker}} = 156.25 - 91.125
\]
Calculating this:
\[
156.25 - 91.125 = 65.125
\]
Therefore, the amount of empty space remaining in the container after the speaker is placed inside is:
\[
\text{Empty space} = 65.125 \text{ cubic inches}
\]
To summarize the answers:
1. Volume of the speaker: **91.125 cubic inches**
2. Volume of the container: **156.25 cubic inches**
3. Empty space remaining in the container: **65.125 cubic inches**
\[
V = s^3
\]
where \( s \) is the length of an edge of the cube. Given that the edge length is 4.5 inches, we can calculate the volume of the speaker:
\[
V_{\text{speaker}} = (4.5)^3
\]
Calculating \( 4.5^3 \):
\[
4.5^3 = 4.5 \times 4.5 \times 4.5
\]
Calculating \( 4.5 \times 4.5 = 20.25 \):
\[
20.25 \times 4.5 = 91.125
\]
Therefore, the volume of the speaker is:
\[
V_{\text{speaker}} = 91.125 \text{ cubic inches}
\]
Next, we calculate the volume of the container, which is in the shape of a square prism. The formula for the volume of a prism is given by:
\[
V = B \times h
\]
where \( B \) is the area of the base and \( h \) is the height. The base area \( B \) is 25 square inches and the height \( h \) is 6.25 inches. Therefore:
\[
V_{\text{container}} = 25 \times 6.25
\]
Calculating this gives:
\[
25 \times 6.25 = 156.25
\]
Thus, the volume of the container is:
\[
V_{\text{container}} = 156.25 \text{ cubic inches}
\]
Now, to find out how much empty space remains in the container after placing the speaker inside, we subtract the volume of the speaker from the volume of the container:
\[
\text{Empty space} = V_{\text{container}} - V_{\text{speaker}} = 156.25 - 91.125
\]
Calculating this:
\[
156.25 - 91.125 = 65.125
\]
Therefore, the amount of empty space remaining in the container after the speaker is placed inside is:
\[
\text{Empty space} = 65.125 \text{ cubic inches}
\]
To summarize the answers:
1. Volume of the speaker: **91.125 cubic inches**
2. Volume of the container: **156.25 cubic inches**
3. Empty space remaining in the container: **65.125 cubic inches**
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