Question

a cereal company is designing containers for a new type of cereal. each container will be shaped like either a rectangular prism or a cylinder and needs to have a volume of 258 3/4 cubic inches.
part a: in one design being considered for the containers shaped like a rectangular prism, each container will have a height of 11 1/2 inches and length 7 1/2 inches. what is the width, in inches, of the container?
part b: in one design being considered for the containers shaped like a cylinder, the container will have a height of 13 inches. what will be the radius of the container, to the nearest tenth of an inch?

Explanation:

Part A

Step1: Convert mixed numbers to improper fractions

The volume formula for a rectangular prism is \(V = l\times w\times h\). Given \(V = 258\frac{3}{4}=\frac{258\times4 + 3}{4}=\frac{1032+3}{4}=\frac{1035}{4}\), \(h = 11\frac{1}{2}=\frac{11\times2+1}{2}=\frac{23}{2}\), \(l = 7\frac{1}{2}=\frac{7\times2 + 1}{2}=\frac{15}{2}\).

Step2: Solve for \(w\)

We know \(V=l\times w\times h\), so \(w=\frac{V}{l\times h}\). Substitute the values: \(l\times h=\frac{15}{2}\times\frac{23}{2}=\frac{345}{4}\). Then \(w=\frac{\frac{1035}{4}}{\frac{345}{4}}\). Since \(\frac{a/b}{c/b}=\frac{a}{c}\) (where \(b
eq0\)), \(w = 3\) inches.

Part B

Step1: Use the cylinder volume formula

The volume formula for a cylinder is \(V=\pi r^{2}h\). Given \(V = 258\frac{3}{4}=\frac{1035}{4}\), \(h = 13\), and \(\pi\approx3.14\).

Step2: Solve for \(r\)

We have \(\frac{1035}{4}=3.14\times r^{2}\times13\). First, calculate \(3.14\times13 = 40.82\). Then \(r^{2}=\frac{1035}{4\times40.82}\). \(\frac{1035}{163.28}\approx6.34\). So \(r=\sqrt{6.34}\approx 2.5\) inches.

Answer:

Part A: \(3\) inches.
Part B: \(2.5\) inches.