10. The volume of a right circular cylinder is \( 252 \mathrm{~cm}^{3} \) and the radius of the base is 4 cm . Find is the height of the cylinder.

To find the height of a right circular cylinder, you can use the formula for the volume of the cylinder, which is given by: \[ V = \pi r^2 h \] Where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height. Given that \( V = 252 \, \text{cm}^3 \) and \( r = 4 \, \text{cm} \), you can rearrange the formula to solve for \( h \): 1. Calculate the base area: \[ \text{Area} = \pi r^2 = \pi (4)^2 = 16\pi \, \text{cm}^2 \] 2. Now, replace \( V \) and the area in the volume formula: \[ 252 = 16\pi h \] 3. Solving for \( h \): \[ h = \frac{252}{16\pi} \approx 5.02 \, \text{cm} \] So, the height of the cylinder is approximately **5.02 cm**. Now, let’s add a couple of extra fun facts: The volume of cylinders has been utilized since ancient times; the Greeks used it in various architectural designs, notably in columns and bases! Who would have thought that math could support gorgeous structures? Also, when working with the volume of cylinders, avoid mixing up the radius and the diameter! It can lead to some seriously distorted results—imagine a towering giant or a pancake flat one!