Question

the volume of rectangular prism is 166 3/5 square inches. the length of the cube is 3 1/5 inches and the width is 8 inches. what is the height of the prism?

Explanation:

Step1: Convert mixed numbers to improper fractions

The volume formula for a rectangular prism is \(V = l\times w\times h\). Given \(V = 166\frac{3}{5}=\frac{166\times5 + 3}{5}=\frac{830+3}{5}=\frac{833}{5}\), \(l = 3\frac{1}{5}=\frac{3\times5+1}{5}=\frac{16}{5}\), \(w = 8\).

Step2: Solve for \(h\)

We know \(h=\frac{V}{l\times w}\). Substitute the values: \(l\times w=\frac{16}{5}\times8=\frac{128}{5}\). Then \(h=\frac{\frac{833}{5}}{\frac{128}{5}}\).
Since \(\frac{a/b}{c/d}=\frac{a}{b}\times\frac{d}{c}\), \(h=\frac{833}{5}\times\frac{5}{128}\). The \(5\)s cancel out, and \(h=\frac{833}{128}=6\frac{65}{128}\)

Answer:

\(6\frac{65}{128}\) inches