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 2 in.​, by how much does the volume​ increase?
7 and one half in.
8 and one fourth in.
4 and one half in.
         ​ (The figure is not to​ scale.)
Question content area bottom
Part 1
The volume of the jewelry box is
  
278 and seven sixteenths
in.cubed.
Part 2
The volume increases by
  
enter your response here in.cubed.

Answer 1

To solve the problem, we'll first calculate the initial volume of the jewelry box using the formula for the volume of a rectangular prism:

\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]

Assuming from your provided dimensions:

- Height = \(4.5\) in = \( \frac{9}{2} \) in
- Length = \(7.5\) in = \( \frac{15}{2} \) in
- Width = \(8.25\) in = \( \frac{33}{4} \) in

Next, we compute the volume step-by-step:

1. **Convert height**:
\[
\text{Height} = 4.5\, \text{in} = \frac{9}{2}\,\text{in}
\]

2. **Convert length**:
\[
\text{Length} = 7.5\, \text{in} = \frac{15}{2}\,\text{in}
\]

3. **Convert width**:
\[
\text{Width} = 8.25\, \text{in} = \frac{33}{4}\,\text{in}
\]

Now let's compute the volume:

\[
\text{Volume} = \left( \frac{15}{2} \right) \times \left( \frac{33}{4} \right) \times \left( \frac{9}{2} \right)
\]

Calculate step by step:

- First, compute the product of \(15\), \(33\), and \(9\):
\[
15 \times 33 = 495
\]

Then, \(495 \times 9 = 4455\)

- Next, compute the product of the denominators:
\[
2 \times 4 \times 2 = 16
\]

Thus, the volume is:
\[
\text{Volume} = \frac{4455}{16}
\]

Now, let's convert this into a decimal or mixed number:
\[
4455 \div 16 = 278.4375 \text{ in}^3
\]

Converting \(278.4375\) to a mixed number gives:
\[
278 + \frac{7}{16} = 278 \frac{7}{16}
\]

So, the volume of the jewelry box is confirmed as:
\[
\text{Volume} = 278 \frac{7}{16} \text{ in}^3
\]

### Part 2: Volume Increase When Height is Increased by 2 in

When the height is increased by 2 in, the new height becomes:
\[
\text{New height} = 4.5 + 2 = 6.5\, \text{in}
\]

Now recalculate the volume with the new height:
\[
\text{New Volume} = \left( \frac{15}{2} \right) \times \left( \frac{33}{4} \right) \times 6.5
\]

Convert \(6.5\) into a fraction:
\[
6.5 = \frac{13}{2}
\]

Now calculate the new volume:
\[
\text{New Volume} = \left( \frac{15}{2} \right) \times \left( \frac{33}{4} \right) \times \left( \frac{13}{2} \right)
\]

Calculate the new product:
- Numerator: \(15 \times 33 \times 13 = 15 \times 429 = 6435\)
- Denominator: \(2 \times 4 \times 2 = 16\)

So:
\[
\text{New Volume} = \frac{6435}{16}
\]

Now calculating the new volume:
\[
6435 \div 16 = 402.1875 \text{ in}^3 = 402 \frac{3}{16}
\]

### Volume Increase:
Now find the increase in volume:
\[
\text{Volume Increase} = \text{New Volume} - \text{Old Volume} = \frac{6435}{16} - \frac{4455}{16} = \frac{1980}{16} = 123.75 \text{ in}^3
\]

Expressed as mixed number:
\[
123.75 = 123 + \frac{3}{4}
\]

So the volume increases by:
\[
\text{Volume Increase} = 123 \frac{3}{4} \text{ in}^3
\]

### Final Outputs:
1. The volume of the jewelry box is \(278 \frac{7}{16} \text{ in}^3\).
2. The volume increase is \(123 \frac{3}{4} \text{ in}^3\).