3.8 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.
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.
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:
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
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