Question

the dimensions of the box below are reduced by half. what is the constant of proportionality between the volume of the new box and the volume of the original box? image of a rectangular prism with dimensions 8 inches, 20 inches, 40 inches multiple choice options: 1/2, 1/4, 1/8, 1/16

Explanation:

Step1: Calculate the original volume

The volume formula for a rectangular prism is \(V = l\times w\times h\). For the original box, \(l = 40\) inches, \(w=20\) inches, \(h = 8\) inches. So \(V_{original}=40\times20\times8=6400\) cubic inches.

Step2: Calculate the new dimensions

Each dimension is reduced by half. New \(l=\frac{40}{2} = 20\) inches, new \(w=\frac{20}{2}=10\) inches, new \(h=\frac{8}{2} = 4\) inches.

Step3: Calculate the new volume

Using the volume formula \(V = l\times w\times h\), \(V_{new}=20\times10\times4 = 800\) cubic inches.

Step4: Find the constant of proportionality

Let the constant of proportionality be \(k\). We know \(V_{new}=k\times V_{original}\). Substitute \(V_{new}=800\) and \(V_{original} = 6400\). Then \(k=\frac{V_{new}}{V_{original}}=\frac{800}{6400}=\frac{1}{8}\)

Answer:

\(\frac{1}{8}\)