Chapter 3.8 Exercise 31

Back to MAT 194F

Chapter 3.8 Exercise 31

Problem

A rope 32 feet long is attached to a weight and passed over a pulley 16 feet above the ground. The other end of the rope is pulled away along the ground at the rate of 3 feet per second. At what rate is the angle between the rope and the ground changing at the instant when the weight is exactly 4 feet off the ground?

Solution

First name our variables:

theta := angle between rope and ground in radians
h := height of weight in feet
x := distance in feet of other end of rope from point on ground below weight
t := elapsed time in seconds, starting when weight is just being lifted from ground.

Then express the mathematical relations among them:

tan theta = x / 16
x = 3 * t
x**2 + 16**2 = (16 + h)**2

Finally calculate the desired value:

d theta / dt = (d theta/dx) * (dx/dt) = (1/(16*sec**2 theta)) * 3
= (3/16) cos**2 theta

If h = 4, then x = 12, tan theta = 3/4 and cos theta = 4/5. The angle in question is therefore increasing at a rate of (3/16)*(4/5)**2 = 3/25 radians per second.