MCV4URelated Rates Problems Note

Often it is required that we find the rate at which one variable is changing, given the rate of change of a related variable.

Steps for Solving a Related Rate Problem

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Related Rate Problem involving a circular model

Eg.1:A pebble is thrown into a lake and causes a circular ripple to spread outward at a rate of 2 m/s. Find the rate of change of the area in terms of π.

a) 3 seconds after the pebble is thrown.

b) When the area of the ripple is 9π m2

Solution:

  1. Make a sketch.
  1. Let Statements:
  1. Required to Find:
  1. Equation that relates the variables.
  1. Implicitly differentiate both sides w.r.t. time.
  1. Substitute in known quantities and known rates.
  1. Solve the equation:
  1. Concluding statement.

b)

Therefore the area of the circle is

Related Rate Problem involving a Right Triangle Model

Eg.2: A dog runs across a path at a rate of 4 m/s. A culvert is located directly beneath the dog. A fish swims right underneath the dog at a rate of 3 m/s. in a direction perpendicular to the path of the dog. Find the rate of change of the distance between the two animals 1 second later.

Solution:

  1. Sketch
  1. Let x represent the distance traveled by the fish (m)

Let y represent the distance traveled by the dog (m)

Let r represent the distance between the dog and the fish (m)

  1. Required to find:
  1. Equation:
  1. Implicitly Differentiate w.r.t. time:
  1. Substitute in known quantities and known rates.
  1. Solve the equation:
  1. Concluding Statement:

Related Rate Problem involving a Conical Model

Eg.3:Water is pouring into an inverted right circular cone at a rate of π m3/min. The height and the diameter of the base of the cone are both 10 m. How fast is the water level rising when the depth of the water is 8 m?

Solution:

  1. Sketch:
  1. Let V represent the volume of water in the cone at time, t.

Let r represent the radius of water in the cone at time, t.

Let h represent the height of water in the cone at time, t.

  1. Required to find:
  1. Equation:
  1. Before we differentiate implicitly we need to have our equation in terms of only h. Using similar triangles from our sketch;
  1. Substitute in known quantities and known rates.
  1. Solve the equation:
  1. Concluding Statement:

Related Rate Problem involving Similar Triangle Model

Eg.4:A student who is 1.6 m tall walks directly away from a lamppost at a rate of 1.2 m/s. A light is situated 8 m above the ground on the lamppost. Show that the student’s shadow is lengthening at a rate of 0.3 m/s. when she is 20 m from the base of the lamppost.

Solution:

  1. Sketch:
  1. Let x represent the length of the shadow.

Let y represent the distance the student is from the base of the lamppost.

  1. We are given: and are required to find:
  1. To determine an equation that relates x and y, we use similar triangles.
  1. Implicitly Differentiate both sides w.r.t. time:
  1. Substitute in known quantities.
  1. Solve the equation:
  1. Concluding Statement:

Related Rate Problem involving a Rectangular Prism Model

Eg.5: The base of a rectangular tank is 3 m by 4 m and is 10 m high. Water is added at a rate of 8 m3/min. Find the rate of change of water level when the water is 5 m deep.

Solution:

  1. Sketch a rectangular prism and label.
  1. Let V represent the volume of the tank.

Let l represent the length of the tank.

Let w represent the width of the tank.

Let h represent the height of the tank.

  1. We are given the length, width and height of the tank and

Required to find when the water is 5 m deep.

  1. Equation that relates the variables.

Length and Width remain constant.

  1. Implicitly Differentiate both sides w.r.t. time.
  1. Substitute in known quantities.
  1. Solve the equation:
  1. Therefore the rate of change of water in the tank is m/min.

Homework: Related Rates Worksheet

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