Question

question 4
a company makes a block of cheese in the shape of a rectangular prism with dimensions 4 inches by 2 inches by 2 inches. the company wants to make a new block, in the shape of a rectangular pyramid, that uses the same amount of cheese. determine two sets of possible dimensions for the pyramid
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Explanation:

Step1: Calculate the volume of the rectangular prism

The volume formula for a rectangular prism is \(V = l\times w\times h\). Given \(l = 4\) inches, \(w=2\) inches, \(h = 2\) inches.

$$V=4\times2\times2=16\space\text{in}^3$$

Step2: Recall the volume formula for a rectangular pyramid

The volume formula for a rectangular pyramid is \(V=\frac{1}{3}l\times w\times h\). Since the volume of the pyramid is equal to the volume of the prism (\(V = 16\space\text{in}^3\)), we have \(\frac{1}{3}lwh=16\), or \(lwh = 48\)

Step3: Find sets of dimensions

• Set 1:

Let \(l = 6\) inches, \(w = 4\) inches. Then from \(lwh=48\), we can solve for \(h\).

$$h=\frac{48}{l\times w}=\frac{48}{6\times4} = 2\space\text{inches}$$

• Set 2:

Let \(l = 8\) inches, \(w = 3\) inches. Then from \(lwh = 48\), we solve for \(h\).

$$h=\frac{48}{l\times w}=\frac{48}{8\times3}=2\space\text{inches}$$

Answer:

One set of dimensions: length \(=6\) inches, width \(=4\) inches, height \(=2\) inches. Another set of dimensions: length \(=8\) inches, width \(=3\) inches, height \(=2\) inches. (Note: There are other possible sets, for example, \(l = 12\), \(w=2\), \(h = 2\) since \(12\times2\times2=48\))