Flashcards in 7.4.2 The Ladder Problem Deck (6):
The Ladder Problem
• 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.
- 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.
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)
100 units / second
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?
1.7 feet / second
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?