Chapter P, Problem 18E

When the Radius Increases

a. A rope is wrapped tightly around a wheel with radius of $2$ feet. If the radius of the wheel is increased by 1 foot to a radius 3 feet, by how much must the rope be lengthened to fit around the wheel?

b. Consider a rope wrapped around the Earth’s equator. We noted in Exercise 17 that the radius of the Earth is about $4000$ miles. That is $21,120,000$ feet. Suppose now that rope is to be suspended exactly 1 foot above the equator. By how much must the rope be lengthened to accomplish this?

The Size of the Earth The radius of the Earth is approximately $4000$ miles.

a. How far is it around the equator? (Hint: You are looking for the circumference of a circle)

b. What is the volume of the Earth? (Hint: The volume of a sphere of radius $r$ is given by $\frac{4}{3}\pi r^3$.)

b. What is the surface area of the Earth? (Hint: The surface area of a sphere of radius $r$ is given by $4\pi r^2$.)