Question
Challenge A jewelry box has the shape of a rectangular prism. Find the volume of the jewelry box. If the height is increased by 3 in., by how much does the volume increase?
6 and one half in.
9 and one fourth in.
5 and one half in.
(The figure is not to scale.)
Question content area bottom
Part 1
The volume of the jewelry box is
330 StartFraction 11 Over 16 EndFraction
in.cubed.
Part 2
The volume increases by
enter your response here in.cubed.
6 and one half in.
9 and one fourth in.
5 and one half in.
(The figure is not to scale.)
Question content area bottom
Part 1
The volume of the jewelry box is
330 StartFraction 11 Over 16 EndFraction
in.cubed.
Part 2
The volume increases by
enter your response here in.cubed.
Answers
GPT-4o mini
To find the volume of the jewelry box, we will use the formula for the volume of a rectangular prism:
\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]
Given the dimensions:
- Height: \( 6.5 \, \text{in.} \) (which is \( 6 \frac{1}{2} = 6.5 \))
- Width: \( 9.25 \, \text{in.} \) (which is \( 9 \frac{1}{4} = 9.25 \))
- Length: \( 5.5 \, \text{in.} \) (which is \( 5 \frac{1}{2} = 5.5 \))
Calculating the volume:
\[
\text{Volume} = 6.5 \times 9.25 \times 5.5
\]
Calculating step by step:
1. Calculate \( 6.5 \times 9.25 \):
\[
6.5 \times 9.25 = 60.125 \, \text{in.}^2
\]
2. Now, multiply this result by \( 5.5 \):
\[
60.125 \times 5.5 = 330.6875 \, \text{in.}^3
\]
To express \( 330.6875 \) in mixed number form, we find the fractional part:
- \( 0.6875 = \frac{11}{16} \)
So, the volume is:
\[
\text{Volume} = 330 \frac{11}{16} \, \text{in.}^3
\]
### Part 1
The volume of the jewelry box is:
\[
330 \frac{11}{16} \, \text{in.}^3
\]
### Part 2
Now, we need to find how much the volume increases when the height is increased by \( 3 \, \text{in.} \).
New height:
\[
6.5 + 3 = 9.5 \, \text{in.}
\]
Calculating the new volume:
\[
\text{New Volume} = 5.5 \times 9.25 \times 9.5
\]
Calculating step by step:
1. Calculate \( 5.5 \times 9.25 \):
\[
5.5 \times 9.25 = 50.875 \, \text{in.}^2
\]
2. Now, multiply this result by \( 9.5 \):
\[
50.875 \times 9.5 = 483.3125 \, \text{in.}^3
\]
Now, express \( 483.3125 \) in mixed number form:
- \( 0.3125 = \frac{5}{16} \)
So, the new volume is:
\[
\text{New Volume} = 483 \frac{5}{16} \, \text{in.}^3
\]
### Calculating Volume Increase
To find the increase in volume:
\[
\text{Increase} = \text{New Volume} - \text{Original Volume} = 483.3125 - 330.6875
\]
Calculating the difference:
\[
\text{Increase} = 152.625
\]
To express \( 152.625 \) in mixed number form:
- \( 0.625 = \frac{5}{8} \)
So, the volume increase is:
\[
\text{Increase} = 152 \frac{5}{8} \, \text{in.}^3
\]
Thus, the answers are:
### Final Answers
1. The volume of the jewelry box is \( 330 \frac{11}{16} \, \text{in.}^3 \).
2. The volume increases by \( 152 \frac{5}{8} \, \text{in.}^3 \).
\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]
Given the dimensions:
- Height: \( 6.5 \, \text{in.} \) (which is \( 6 \frac{1}{2} = 6.5 \))
- Width: \( 9.25 \, \text{in.} \) (which is \( 9 \frac{1}{4} = 9.25 \))
- Length: \( 5.5 \, \text{in.} \) (which is \( 5 \frac{1}{2} = 5.5 \))
Calculating the volume:
\[
\text{Volume} = 6.5 \times 9.25 \times 5.5
\]
Calculating step by step:
1. Calculate \( 6.5 \times 9.25 \):
\[
6.5 \times 9.25 = 60.125 \, \text{in.}^2
\]
2. Now, multiply this result by \( 5.5 \):
\[
60.125 \times 5.5 = 330.6875 \, \text{in.}^3
\]
To express \( 330.6875 \) in mixed number form, we find the fractional part:
- \( 0.6875 = \frac{11}{16} \)
So, the volume is:
\[
\text{Volume} = 330 \frac{11}{16} \, \text{in.}^3
\]
### Part 1
The volume of the jewelry box is:
\[
330 \frac{11}{16} \, \text{in.}^3
\]
### Part 2
Now, we need to find how much the volume increases when the height is increased by \( 3 \, \text{in.} \).
New height:
\[
6.5 + 3 = 9.5 \, \text{in.}
\]
Calculating the new volume:
\[
\text{New Volume} = 5.5 \times 9.25 \times 9.5
\]
Calculating step by step:
1. Calculate \( 5.5 \times 9.25 \):
\[
5.5 \times 9.25 = 50.875 \, \text{in.}^2
\]
2. Now, multiply this result by \( 9.5 \):
\[
50.875 \times 9.5 = 483.3125 \, \text{in.}^3
\]
Now, express \( 483.3125 \) in mixed number form:
- \( 0.3125 = \frac{5}{16} \)
So, the new volume is:
\[
\text{New Volume} = 483 \frac{5}{16} \, \text{in.}^3
\]
### Calculating Volume Increase
To find the increase in volume:
\[
\text{Increase} = \text{New Volume} - \text{Original Volume} = 483.3125 - 330.6875
\]
Calculating the difference:
\[
\text{Increase} = 152.625
\]
To express \( 152.625 \) in mixed number form:
- \( 0.625 = \frac{5}{8} \)
So, the volume increase is:
\[
\text{Increase} = 152 \frac{5}{8} \, \text{in.}^3
\]
Thus, the answers are:
### Final Answers
1. The volume of the jewelry box is \( 330 \frac{11}{16} \, \text{in.}^3 \).
2. The volume increases by \( 152 \frac{5}{8} \, \text{in.}^3 \).