Question

1. if the volume of a rectangular prism is doubled, what can be said about its dimensions?

a. one or more dimensions must increase.
b. the width must remain constant.
c. the length must be doubled.
d. the height must be halved.

Explanation:

Step1: Recall the volume formula

The volume \(V\) of a rectangular prism is \(V = l\times w\times h\) (where \(l\) is length, \(w\) is width, \(h\) is height).

Step2: Analyze option a

If \(V\) is doubled (\(V_{new}=2V\)), assume \(l\times w\times h\) becomes \(l_{new}\times w_{new}\times h_{new}\). If \(V_{new} = 2(l\times w\times h)\), then at least one of \(l_{new}\gt l\), \(w_{new}\gt w\), or \(h_{new}\gt h\) must be true. For example, if \(l_{new}=2l\), \(w_{new} = w\), \(h_{new}=h\), \(V_{new}=2l\times w\times h = 2V\); or if \(l_{new}=1.5l\), \(w_{new}=1.5w\), \(h_{new}=h\), \(V_{new}=1.5l\times1.5w\times h=2.25V\gt2V\) (a case where two dimensions increase).

Step3: Analyze option b

If the width remains constant (\(w_{new}=w\)), if \(l_{new}=l\) and \(h_{new}=h\), \(V_{new}=V\). If we want \(V_{new} = 2V\), \(w\) does not have to be constant. For example, \(l_{new}=2l\), \(w_{new}=1.5w\), \(h_{new}=h\), \(V_{new}=3V\) (width is not constant).

Step4: Analyze option c

If only the length is doubled (\(l_{new} = 2l\), \(w_{new}=w\), \(h_{new}=h\)), \(V_{new}=2V\). But there are other ways. For example, \(w_{new}=2w\), \(l_{new}=l\), \(h_{new}=h\) also gives \(V_{new}=2V\). So length doubling is not the only way.

Step5: Analyze option d

If the height is halved (\(h_{new}=\frac{1}{2}h\)), if \(l_{new}=l\) and \(w_{new}=w\), \(V_{new}=\frac{1}{2}V\), which is the opposite of what we want.

Answer:

A. One or more dimensions must increase.