Question

1. a conical silo has a volume of $500\pi\mathrm{m}^{3}$ and a radius of 5 m. what is the height of the silo?

a. 55 m
b. 30 m
c. 42 m
d. 60 m

Explanation:

Step1: Recall the volume formula for a cone

The volume formula for a cone is \(V=\frac{1}{3}\pi r^{2}h\), where \(V\) is the volume, \(r\) is the radius, and \(h\) is the height.

Step2: Substitute the given values into the formula

We are given that \(V = 500\pi\) and \(r = 5\). Substituting these into the formula gives \(500\pi=\frac{1}{3}\pi(5)^{2}h\).

Step3: Simplify the equation

First, simplify \(\frac{1}{3}\pi(5)^{2}h\) to \(\frac{25\pi}{3}h\). So the equation becomes \(500\pi=\frac{25\pi}{3}h\).

Step4: Solve for \(h\)

Divide both sides of the equation by \(\pi\) (since \(\pi
eq0\)), we get \(500=\frac{25}{3}h\). Then multiply both sides by \(\frac{3}{25}\), \(h = 500\times\frac{3}{25}=60\). Wait, no, let's correct:
From \(500\pi=\frac{1}{3}\pi\times25h\), cancel \(\pi\) (since \(\pi
eq0\)), we have \(500=\frac{25}{3}h\). Cross - multiply: \(h=\frac{500\times3}{25}= 60\div2 = 30\) (because \(500\times3 = 1500\), \(1500\div25=60\), but wait the formula is \(V=\frac{1}{3}\pi r^{2}h\), so \(h=\frac{3V}{\pi r^{2}}\). Substitute \(V = 500\pi\), \(r = 5\): \(h=\frac{3\times500\pi}{\pi\times25}=\frac{1500}{25}=60\div2 = 30\) (because \(3\times500 = 1500\), \(1500\div25 = 60\), but actually \(h=\frac{3V}{\pi r^{2}}=\frac{3\times500\pi}{\pi\times25}=60\div2=30\)).

So the height \(h = 30\) m.

Answer:

B. 30 m