To find the surface area of a rectangular prism, we can use the formula:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height of the prism.
Given:
- Length (\( l \)) = \( 3 \frac{3}{4} \) dm = \( \frac{15}{4} \) dm
- Width (\( w \)) = \( 4 \frac{1}{2} \) dm = \( \frac{9}{2} \) dm
- Height (\( h \)) = \( 5 \) dm
Converting all dimensions to fractions:
- \( l = \frac{15}{4} \) dm
- \( w = \frac{9}{2} = \frac{18}{4} \) dm
- \( h = 5 = \frac{20}{4} \) dm
Now we can plug these values into the surface area formula:
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Calculate \( lw \): \[ lw = \left(\frac{15}{4}\right) \left(\frac{18}{4}\right) = \frac{270}{16} = \frac{135}{8} \]
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Calculate \( lh \): \[ lh = \left(\frac{15}{4}\right) \left(\frac{20}{4}\right) = \frac{300}{16} = \frac{75}{4} \]
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Calculate \( wh \): \[ wh = \left(\frac{18}{4}\right) \left(\frac{20}{4}\right) = \frac{360}{16} = \frac{90}{4} \]
Now we can sum \( lw + lh + wh \): \[ lw + lh + wh = \frac{135}{8} + \frac{75}{4} + \frac{90}{4} \]
Next, we convert everything to a common denominator (which is 8):
- \( \frac{75}{4} = \frac{150}{8} \)
- \( \frac{90}{4} = \frac{180}{8} \)
Now we can sum them up: \[ \frac{135}{8} + \frac{150}{8} + \frac{180}{8} = \frac{465}{8} \]
Finally, we multiply by 2 to find the surface area: \[ \text{Surface Area} = 2 \left(\frac{465}{8}\right) = \frac{930}{8} = \frac{465}{4} = 116 \frac{1}{4} \text{ dm}^2 \]
Therefore, the surface area of the rectangular prism is \( 116 \frac{1}{4} \text{ dm}^2 \).
Response: 116 1/4 dm²