7.4.2 The Ladder Problem

Question 1

The Ladder Problem

Answer 1

• Related rate problems involve using a known rate of change to find an associated rate of change.
• Use implicit differentiation when you cannot write the dependent variable in terms of the independent variable.

Question 2

note

Answer 2

• When a ladder slides down a wall, the rate at which it falls downward is not necessarily equal to the rate at which the base of the ladder moves away from the wall.
• Suppose you are given a 13-foot ladder and you are told that the ladder is moving 6 feet per second away from the wall when the ladder is 12 feet away from the wall. What is the rate of change in the y-direction?
• This is another example of a related rate. The rate at which a ladder falls downward depends on the rate at which the ladder moves away from the wall.
• You want to know the rate of change in the y-direction when the ladder is 12 feet from the wall.
• You are told that the rate of change in the x-direction when the ladder is 12 feet from the wall is 6 feet per second.
• Because the Pythagorean theorem relates the distance away from the wall to the distance to the top of the ladder, the rates of change can be related by taking a derivative.
• Notice that you can express the derivative of y with respect to time in terms of the derivative of x with respect to time, the x-value, and the y-value.
• Substitute to find the value of dy/dt.

Question 3

Suppose a particle is moving from left to right along the graph of y = x^ 2. Find the rate of change of the distance between the particle and the origin at the instant x = 5 if the particle moves horizontally at a constant rate of 10 units / second. (In other words, dx / dt = 10)

Answer 3

100 units / second

Question 4

A winch on a motionless truck 6 feet above the ground is dragging a heavy load (see diagram).
If the winch pulls the cable at a constant rate of 1.5 feet / second, how quickly is the load moving on the ground when it is 11 feet from the truck?

Answer 4

1.7 feet / second

Question 5

Jim, who is 6 ft tall, is walking directly away from a 15 ft lamppost at a rate of 4 ft per sec. What is the rate of change in the length of Jim’s shadow when he is 8 ft from the base of the lamppost?

Answer 5

8/3 feet/second

Question 6

A 10 ft ladder is being pulled away from
a wall at a rate of 3 ft/sec. What is the rate
of change in the area beneath the ladder
when the ladder is 6 ft from the wall?

Answer 6

dA/dt=214 ft^2/sec