To find the surface area of a rectangular prism, we use the formula:
\[ \text{Surface Area} = 2lw + 2lh + 2wh \]
where \(l\) is the length, \(w\) is the width, and \(h\) is the height.
Given the areas of the three pairs of opposite faces of the rectangular prism:
- First face: \(lw = 15\) in²
- Second face: \(lh = 25\) in²
- Third face: \(wh = 45\) in²
We can denote:
\[ lw = 15 \] \[ lh = 25 \] \[ wh = 45 \]
We need to find \(l\), \(w\), and \(h\). We can find \(l\), \(w\), and \(h\) using these equations.
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From \(lw = 15\), we can express \(w\) in terms of \(l\): \[ w = \frac{15}{l} \]
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Substitute \(w\) into the equation \(wh = 45\): \[ \frac{15}{l}h = 45 \] \[ h = \frac{45l}{15} = 3l \]
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Substitute \(h\) into the equation \(lh = 25\): \[ l(3l) = 25 \] \[ 3l^2 = 25 \] \[ l^2 = \frac{25}{3} \] \[ l = \sqrt{\frac{25}{3}} = \frac{5}{\sqrt{3}} \approx 2.89 \text{ in} \]
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Now we can find \(h\): \[ h = 3l = 3 \left( \frac{5}{\sqrt{3}} \right) = \frac{15}{\sqrt{3}} \approx 8.66 \text{ in} \]
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Now we can find \(w\): \[ w = \frac{15}{l} = \frac{15}{\frac{5}{\sqrt{3}}} = 3\sqrt{3} \approx 5.20 \text{ in} \]
Now we have all dimensions:
- \(l = \frac{5}{\sqrt{3}}\)
- \(w = 3\sqrt{3}\)
- \(h = \frac{15}{\sqrt{3}}\)
Now we can calculate the surface area: \[ \text{Surface Area} = 2lw + 2lh + 2wh \]
Calculate each term:
- \(2lw = 2 \left(\frac{5}{\sqrt{3}}\right) (3\sqrt{3}) = 2 \cdot 5 \cdot \sqrt{3} = 30\)
- \(2lh = 2 \left( \frac{5}{\sqrt{3}} \right) \left( \frac{15}{\sqrt{3}} \right) = 2 \cdot \frac{75}{3} = 50\)
- \(2wh = 2 (3\sqrt{3}) \left( \frac{15}{\sqrt{3}} \right) = 2 (45) = 90\)
Now combine these: \[ \text{Surface Area} = 30 + 50 + 90 = 170 \text{ in}^2 \]
Thus, the surface area of the rectangular prism is 170 in².