Question

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

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

Explanation:

The volume formula for a rectangular prism is \(V = l\times w\times h\) (where \(l\) is length, \(w\) is width, and \(h\) is height). If the volume is doubled (\(V_{new}=2V\)), we can analyze each option:

• Option a: If the height is halved (\(h_{new}=\frac{h}{2}\)), then \(V_{new}=l\times w\times\frac{h}{2}=\frac{V}{2}\), which is incorrect.
• Option b: Since \(V = l\times w\times h\), if \(V_{new} = 2V\), then by the property of multiplication, one or more of \(l\), \(w\), or \(h\) must increase. For example, if \(l_{new}=2l\), \(V_{new}=2l\times w\times h = 2V\); if \(w_{new}=2w\), \(V_{new}=l\times2w\times h = 2V\); if \(h_{new}=2h\), \(V_{new}=l\times w\times2h = 2V\); or if \(l_{new}=1.5l\) and \(w_{new}=1.33w\) (approximate values such that \(1.5\times1.33\approx2\)), \(V_{new}=1.5l\times1.33w\times h\approx2V\).
• Option c: The width does not have to remain constant. For example, if \(w_{new} = 2w\) and \(l\) and \(h\) remain the same, \(V_{new}=2V\).
• Option d: The length does not have to be the one that is doubled. As shown above, the width or height could be the dimension (or a combination of dimensions) that changes to double the volume.

Answer:

B. One or more dimensions must increase.