The height of a cylinder is 4 millimeters. The circumference of its base is \( 10 \pi \mathrm{~mm} \). a. What is the radius of the cylinder? Explain. b. Find the volume of the cylinder to the nearest tenth.

To find the radius of the cylinder, we can use the formula for circumference, which is \( C = 2\pi r \). Given that the circumference is \( 10\pi \mathrm{~mm} \), we can set up the equation: \[ 10\pi = 2\pi r. \] Dividing both sides by \( 2\pi \), we find: \[ r = \frac{10\pi}{2\pi} = 5 \text{ mm}. \] Now, for the volume of the cylinder, we use the formula \( V = \pi r^2 h \). Substituting the radius \( r = 5 \text{ mm} \) and height \( h = 4 \text{ mm} \) into the formula gives us: \[ V = \pi (5)^2 (4) = \pi (25)(4) = 100\pi. \] To find the volume to the nearest tenth, we calculate \( 100\pi \approx 314.2 \text{ mm}^3 \). To keep your cylinder education exciting, let's take a quick detour into the world of historic marvels! Did you know that many ancient civilizations, including the Greeks and Egyptians, utilized cylinders in various forms? From architecture to tools, their properties were fundamental in building some of the most iconic structures we still admire today! Speaking of cylinders, they also play a vital role in science and engineering! For instance, engineers use the understanding of cylindrical shapes to design everything from water towers to rocket engines. Knowing how to calculate the volume helps ensure these structures can hold the necessary materials and withstand environmental pressures.