3.8    Related Rates

Related Rates

Two variables, perhaps x and y, are both functions of a third variable, time, t, and x and y are related by an equation.

Example A fire has started in a dry, open field and spreads in the form of a circle.  The radius of the circle increases at a rate of 6 ft/min.  Find the rate at which the fire area is increasing when the radius is 150 ft.

Strategy:  Draw and label picture.  What are we finding?  Name variables and equations involved.  (Substitute) and differentiate, then "plug-in" values.

Finding:                   Equation:

Implicitly differentiate with respect to t.  Note:  The area and radius are both functions of t.

Given:  ft/min   and  r = 150

Example:   A ladder 26 ft long leans against a vertical wall.  The foot of the ladder is drawn away from the wall at a rate of 4 ft/s.  How fast is the top of the ladder sliding down the wall, when the foot of the ladder is 10 ft from the wall?

Strategy:  Draw and label pictures.  What are we finding?   Name variables and equations involved.  (Substitute) and differentiate, then "plug-in" values.

Find:                         Equation:

Differentiate implicitly with respect to the variable t.  Note:  x and y are both functions of t.

Given:        Must find y using

Example:  Water runs into a conical tank shown at a constant rate of 2 ft3 per minute.  The dimensions of the tank are altitude of 12ft and base radius of 6 ft.  How fast is the water level rising when the water is 6 feet deep?

Draw a picture.  Need both the volume of a cone and similar triangle proportions.

Find:            Volume of cone =                                                      Similar Triangle:

Given:

Example:   A spherical balloon is inflated with gas at the rate of 100 ft3/min.  Assuming the gas pressure remains constant, how fast is the radius of the balloon increasing when the radius is 3 ft?

Find:               Volume of Sphere =

Given:  r = 3        Know:

Example:   A man 6 ft tall walks at the rate of 5 ft/sec. toward a street light that is 16 ft. above the ground.  At what rate is the tip of his shadow moving?

Find:                         Similar Triangles

Given:

Assignment  3.8  pg 186; 1-11 odd, 12 [ans:-1/(20π)], 13,15, 16 [ans: 0.6 m/s]19, 20 [ans: ] 21, 27, 29, 30 [ans: -1/8 rad/s], 31