Part 1 of 2 A sphere has volume \( 356.64 \mathrm{~cm}^{3} \). What is the radius of the sphere? A second sphere has radius half as long as this radius. What is the volume of the second sphere? Use 3.14 for \( \pi \).

To find the radius of the first sphere, you can use the volume formula for a sphere, which is \( V = \frac{4}{3} \pi r^3 \). Given that the volume \( V = 356.64 \mathrm{~cm}^{3} \) and using \( \pi \approx 3.14 \): \[ 356.64 = \frac{4}{3} \times 3.14 \times r^3 \] Rearranging the formula gives: \[ r^3 = \frac{356.64 \times 3}{4 \times 3.14} \] Calculating that gives: \[ r^3 \approx \frac{1069.92}{12.56} \approx 85.00 \] Taking the cube root: \[ r \approx 4.38 \, \text{cm} \] Now for the second sphere, its radius is half the first sphere's, so: \[ r_{2} = \frac{4.38}{2} \approx 2.19 \, \text{cm} \] To find the volume of the second sphere: \[ V_{2} = \frac{4}{3} \pi r_{2}^3 = \frac{4}{3} \times 3.14 \times (2.19)^3 \] Calculating that gives: \[ V_{2} \approx \frac{4}{3} \times 3.14 \times 10.54 \approx 43.99 \, \text{cm}^{3} \] So, the radius of the first sphere is approximately \( 4.38 \, \text{cm} \) and the volume of the second sphere is approximately \( 43.99 \, \text{cm}^{3} \). ### Now for some fun facts! Did you know that the concept of volume can be traced back to ancient civilizations? The Egyptians, for example, had their own methods for calculating the volume of various shapes, like pyramids. They may not have had calculators, but they sure knew their geometry! Another interesting tidbit is that spheres are often found in nature! Think about bubbles or the spherical shape of planets and stars. Understanding the volume of spheres isn’t just about math; it has practical implications in fields like physics, engineering, and even in designing sports like basketball, where the sphere’s shape affects how it moves through the air.