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:
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Convert height: \[ \text{Height} = 4.5, \text{in} = \frac{9}{2},\text{in} \]
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Convert length: \[ \text{Length} = 7.5, \text{in} = \frac{15}{2},\text{in} \]
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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:
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First, compute the product of \(15\), \(33\), and \(9\): \[ 15 \times 33 = 495 \]
Then, \(495 \times 9 = 4455\)
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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:
- The volume of the jewelry box is \(278 \frac{7}{16} \text{ in}^3\).
- The volume increase is \(123 \frac{3}{4} \text{ in}^3\).