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
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
enter your response here in.cubed.
Answers
GPT-4o mini
To find the volume of a rectangular prism (which is the shape of the jewelry box), you can use the formula:
\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]
Given:
- Height = \(6.5 \text{ in} = \frac{13}{2} \text{ in}\)
- Width = \(9.25 \text{ in} = \frac{37}{4} \text{ in}\)
- Length = \(5.5 \text{ in} = \frac{11}{2} \text{ in}\)
Now, let's calculate the volume:
1. Convert all dimensions into fractional form (if not already):
- Length = \( \frac{11}{2} \)
- Width = \( \frac{37}{4} \)
- Height = \( \frac{13}{2} \)
2. Plug these values into the volume formula:
\[
\text{Volume} = \frac{11}{2} \times \frac{37}{4} \times \frac{13}{2}
\]
3. Multiply the fractions together:
\[
\text{Volume} = \frac{11 \times 37 \times 13}{2 \times 4 \times 2}
\]
Calculating the numerator:
\[
11 \times 37 = 407
\]
\[
407 \times 13 = 5291
\]
Calculating the denominator:
\[
2 \times 4 \times 2 = 16
\]
Thus,
\[
\text{Volume} = \frac{5291}{16} \text{ in}^3
\]
4. The volume of the jewelry box is:
\[
\text{Volume} = 330.6875 \text{ in}^3 \quad \text{(approximately)}
\]
**Part 2: Increase in volume if height is increased by 3 inches**
If the height is increased by \(3 \text{ in}\), the new height will be:
\[
\text{New Height} = 6.5 + 3 = 9.5 \text{ in}
\]
Now calculate the new volume:
\[
\text{New Volume} = \text{length} \times \text{width} \times \text{new height}
\]
\[
\text{New Volume} = \frac{11}{2} \times \frac{37}{4} \times 9.5
\]
Convert \(9.5\) into an improper fraction for easier calculation:
\[
9.5 = \frac{19}{2}
\]
So:
\[
\text{New Volume} = \frac{11}{2} \times \frac{37}{4} \times \frac{19}{2}
\]
Now calculate:
1. Multiply:
\[
\text{New Volume} = \frac{11 \times 37 \times 19}{2 \times 4 \times 2}
\]
Calculating the new numerator:
\[
11 \times 37 = 407
\]
\[
407 \times 19 = 7723
\]
Calculating the denominator is the same as before:
\[
2 \times 4 \times 2 = 16
\]
Thus,
\[
\text{New Volume} = \frac{7723}{16} \text{ in}^3
\]
Now, we need to find the increase in volume:
\[
\text{Increase in Volume} = \text{New Volume} - \text{Original Volume}
\]
Substituting the values:
\[
\text{Increase in Volume} = \frac{7723}{16} - \frac{5291}{16} = \frac{7723 - 5291}{16} = \frac{2432}{16} = 152 \text{ in}^3
\]
**Final Results:**
- The volume of the jewelry box is approximately **330.69 in³**.
- The increase in volume, when the height is increased by 3 in., is **152 in³**.
\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]
Given:
- Height = \(6.5 \text{ in} = \frac{13}{2} \text{ in}\)
- Width = \(9.25 \text{ in} = \frac{37}{4} \text{ in}\)
- Length = \(5.5 \text{ in} = \frac{11}{2} \text{ in}\)
Now, let's calculate the volume:
1. Convert all dimensions into fractional form (if not already):
- Length = \( \frac{11}{2} \)
- Width = \( \frac{37}{4} \)
- Height = \( \frac{13}{2} \)
2. Plug these values into the volume formula:
\[
\text{Volume} = \frac{11}{2} \times \frac{37}{4} \times \frac{13}{2}
\]
3. Multiply the fractions together:
\[
\text{Volume} = \frac{11 \times 37 \times 13}{2 \times 4 \times 2}
\]
Calculating the numerator:
\[
11 \times 37 = 407
\]
\[
407 \times 13 = 5291
\]
Calculating the denominator:
\[
2 \times 4 \times 2 = 16
\]
Thus,
\[
\text{Volume} = \frac{5291}{16} \text{ in}^3
\]
4. The volume of the jewelry box is:
\[
\text{Volume} = 330.6875 \text{ in}^3 \quad \text{(approximately)}
\]
**Part 2: Increase in volume if height is increased by 3 inches**
If the height is increased by \(3 \text{ in}\), the new height will be:
\[
\text{New Height} = 6.5 + 3 = 9.5 \text{ in}
\]
Now calculate the new volume:
\[
\text{New Volume} = \text{length} \times \text{width} \times \text{new height}
\]
\[
\text{New Volume} = \frac{11}{2} \times \frac{37}{4} \times 9.5
\]
Convert \(9.5\) into an improper fraction for easier calculation:
\[
9.5 = \frac{19}{2}
\]
So:
\[
\text{New Volume} = \frac{11}{2} \times \frac{37}{4} \times \frac{19}{2}
\]
Now calculate:
1. Multiply:
\[
\text{New Volume} = \frac{11 \times 37 \times 19}{2 \times 4 \times 2}
\]
Calculating the new numerator:
\[
11 \times 37 = 407
\]
\[
407 \times 19 = 7723
\]
Calculating the denominator is the same as before:
\[
2 \times 4 \times 2 = 16
\]
Thus,
\[
\text{New Volume} = \frac{7723}{16} \text{ in}^3
\]
Now, we need to find the increase in volume:
\[
\text{Increase in Volume} = \text{New Volume} - \text{Original Volume}
\]
Substituting the values:
\[
\text{Increase in Volume} = \frac{7723}{16} - \frac{5291}{16} = \frac{7723 - 5291}{16} = \frac{2432}{16} = 152 \text{ in}^3
\]
**Final Results:**
- The volume of the jewelry box is approximately **330.69 in³**.
- The increase in volume, when the height is increased by 3 in., is **152 in³**.