Question

if a point in the cylinder below is chosen at random, what is the probabilitiy that it within the shaded region?

Explanation:

Step1: Calculate the volume of the large cylinder

The formula for the volume of a cylinder is \(V=\pi r^{2}h\). The radius of the large cylinder \(R = \frac{16}{2}=8\) inches and height \(h = 14\) inches. So \(V_{1}=\pi\times8^{2}\times14=\pi\times64\times14 = 896\pi\) cubic inches.

Step2: Calculate the volume of the small cylinder

The radius of the small cylinder \(r=\frac{10}{2} = 5\) inches and height \(h = 14\) inches. So \(V_{2}=\pi\times5^{2}\times14=\pi\times25\times14=350\pi\) cubic inches.

Step3: Calculate the volume of the shaded region

The volume of the shaded region \(V = V_{1}-V_{2}\). Substitute the values of \(V_{1}\) and \(V_{2}\): \(V=(896\pi - 350\pi)=546\pi\) cubic inches.

Step4: Calculate the probability

The probability \(P\) that a randomly - chosen point is in the shaded region is \(P=\frac{V}{V_{1}}\). Substitute \(V = 546\pi\) and \(V_{1}=896\pi\): \(P=\frac{546\pi}{896\pi}=\frac{546}{896}=\frac{39}{64}\)

Answer:

\(\frac{39}{64}\)