Question

terry is filling spherical water balloons with a faucet that puts out $100\\ \mathrm{cm}^3$ of water per second. at that rate, it takes terry 6 seconds to fill a balloon.
what is the radius of a filled balloon?
round to the nearest centimeter.
\boxed{\\ \\ }\\ \mathrm{cm}
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Explanation:

Step1: Calculate the volume of the balloon

The faucet puts out \(100\space cm^3\) per second and it takes 6 seconds to fill the balloon. So the volume \(V\) of the balloon is the rate times time, which is \(V = 100\times6=600\space cm^3\).

Step2: Use the volume formula for a sphere

The volume formula for a sphere is \(V=\frac{4}{3}\pi r^3\), where \(r\) is the radius. We know \(V = 600\), so we can solve for \(r\).
First, rewrite the formula to solve for \(r^3\):

$$r^3=\frac{3V}{4\pi}$$

Substitute \(V = 600\) into the formula:

$$r^3=\frac{3\times600}{4\pi}=\frac{1800}{4\pi}=\frac{450}{\pi}$$

Calculate \(\frac{450}{\pi}\approx\frac{450}{3.1416}\approx143.24\)

Step3: Solve for \(r\)

Take the cube root of \(r^3\) to find \(r\):

$$r=\sqrt[3]{143.24}\approx5.23$$

Round to the nearest centimeter, \(r\approx5\)

Answer:

5