To find the surface area of a rectangular prism, you 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.
First, convert the mixed numbers to improper fractions:
- 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 = \( 5 \) dm (or \( \frac{10}{2} \) for easier calculation with fractions)
Now, substitute these values into the surface area formula:
\[ \text{Surface Area} = 2 \left(\left(\frac{15}{4} \cdot \frac{9}{2}\right) + \left(\frac{15}{4} \cdot 5\right) + \left(\frac{9}{2} \cdot 5\right)\right) \]
Now calculate each term:
- \( lw = \frac{15}{4} \cdot \frac{9}{2} = \frac{15 \cdot 9}{4 \cdot 2} = \frac{135}{8} \)
- \( lh = \frac{15}{4} \cdot 5 = \frac{15 \cdot 5}{4} = \frac{75}{4} \)
- \( wh = \frac{9}{2} \cdot 5 = \frac{9 \cdot 5}{2} = \frac{45}{2} \)
Now, combine these terms:
To add the fractions, we need a common denominator. The least common multiple of 8, 4, and 2 is 8.
Convert each term:
- \( lw = \frac{135}{8} \) (already in terms of 8)
- \( lh = \frac{75}{4} = \frac{75 \cdot 2}{4 \cdot 2} = \frac{150}{8} \)
- \( wh = \frac{45}{2} = \frac{45 \cdot 4}{2 \cdot 4} = \frac{180}{8} \)
Now add them together:
\[ lw + lh + wh = \frac{135}{8} + \frac{150}{8} + \frac{180}{8} = \frac{135 + 150 + 180}{8} = \frac{465}{8} \]
Now multiply by 2 for the surface area:
\[ \text{Surface Area} = 2 \cdot \frac{465}{8} = \frac{930}{8} = 116.25 \]
Thus, the surface area of the rectangular prism is:
\[ \text{Surface Area} = 116.25 \text{ square decimeters} \]