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