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AP Calculus AB: 7.4.2 The Ladder Problem

Mathematics6 CardsCreated 3 months ago

This content explores related rates problems involving implicit differentiation, such as determining how fast the top of a sliding ladder falls given the rate its base moves away from the wall. It also covers finding the rate of change of the distance between a moving particle on a curve and the origin using derivatives.

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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.

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Key Terms

Term
Definition

The Ladder Problem

  • Related rate problems involve using a known rate of change to find an associated rate of change.

  • Use implicit differentiatio...

note

  • 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 ...

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?

8/3 feet/second

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?

dA/dt=214 ft^2/sec

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AP Calculus AB: 7.4.2 The Ladder Problem
TermDefinition

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.

note

  • 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?

8/3 feet/second

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?

dA/dt=214 ft^2/sec