Chapter 1.5, Problem 64E

Numerical and Graphical Reasoning A crossed belt connects a 20–centimeter pulley (10–cm radius) on an electric motor with a 40–centimeter pulley (20–cm radius) on a saw arbor (see figure). The electric motor runs at 1700 revolutions per minute.

(a) Determine the number of revolutions per minute of the saw.

(b) How does crossing the belt affect the saw in relation to the motor?

(c) Let L be the total length of the belt. Write L as a function of $\varphi$. where $\varphi$ is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulley.)

(d) Use a graphing utility to complete the table.

$\varphi$	0.3	0.6	0.9	1.2	1.5
L

(e) Use a graphing utility to graph the function over the appropriate domain.

(f) Find $\underset{\varphi \to (\pi /2)}{\lim }$ L.

(g) Use a geometric argument as the basis of a second method of finding the limit in part (f).

(h) Find $\underset{\varphi \to 0}{\lim }$ L.