Question

what is the volume of the rectangular prism shown? image of a green rectangular prism with dimensions labeled: ( 3\frac{1}{2} ) in., ( 1\frac{7}{8} ) in., ( 2\frac{3}{4} ) in. multiple - choice options: ( 8\frac{1}{8} ) in.³, ( 21\frac{11}{12} ) in.³, ( 6\frac{31}{64} ) in.³, ( 18\frac{3}{64} ) in.³

Explanation:

Step1: Recall volume formula for rectangular prism

The volume \( V \) of a rectangular prism is given by \( V = l \times w \times h \), where \( l \) is length, \( w \) is width, and \( h \) is height. Here, \( l = 3\frac{1}{2} \) in, \( w = 1\frac{7}{8} \) in, \( h = 2\frac{3}{4} \) in.

Step2: Convert mixed numbers to improper fractions

• \( 3\frac{1}{2}=\frac{3\times2 + 1}{2}=\frac{7}{2} \)
• \( 1\frac{7}{8}=\frac{1\times8 + 7}{8}=\frac{15}{8} \)
• \( 2\frac{3}{4}=\frac{2\times4 + 3}{4}=\frac{11}{4} \)

Step3: Multiply the fractions

\( V=\frac{7}{2}\times\frac{15}{8}\times\frac{11}{4} \)
First, multiply \( \frac{7}{2}\times\frac{15}{8}=\frac{105}{16} \)
Then, multiply \( \frac{105}{16}\times\frac{11}{4}=\frac{1155}{64} \)

Step4: Convert improper fraction to mixed number

\( \frac{1155}{64}=18\frac{3}{64} \) (since \( 64\times18 = 1152 \), \( 1155 - 1152 = 3 \))

Answer:

\( 18\frac{3}{64}\text{ in}^3 \)