Question

teresas company makes solid balls out of scrap metal for various industrial uses. her current task is to make a batch of aluminum balls with radius 4.5 in. if she has a total of 76,302 in³ of aluminum, how many balls can she make? use 3.14 for π, and do not round your answer.

Explanation:

Step1: Calculate the volume of one ball

The volume formula for a sphere is \(V = \frac{4}{3}\pi r^{3}\). Given \(r = 4.5\) in and \(\pi=3.14\).
Substitute the values into the formula:
\(V=\frac{4}{3}\times3.14\times(4.5)^{3}\)
First, calculate \((4.5)^{3}=4.5\times4.5\times4.5 = 91.125\)
Then, \(\frac{4}{3}\times3.14\times91.125=4\times3.14\times30.375\) (since \(\frac{91.125}{3}=30.375\))
\(4\times3.14\times30.375 = 381.51\) \(in^{3}\)

Step2: Calculate the number of balls

Let \(n\) be the number of balls. We know the total volume of aluminum is \(V_{total}=76302\) \(in^{3}\)
The formula for \(n\) is \(n=\frac{V_{total}}{V}\)
Substitute \(V_{total} = 76302\) \(in^{3}\) and \(V = 381.51\) \(in^{3}\)
\(n=\frac{76302}{381.51}=200\)

Answer:

200