Related Rates
1. Read the problem and draw a picture (if necessary). Label all
pieces that are constant.
2. Identify what you have been given. Make sure you know the
difference between quantities (lengths) and speeds (derivatives).
3. Identify a formula to represent the problem. The formula is often
geometric (Ex. Area, Volume, SOHCAHTOA, Pythagorean
Theorem, etc.). If you have variables in your equation that are not
mentioned in the problem, you will need to change your equation to
only be in terms of one variable (using proportions).
4. Plug in any values that are constant (not changing with respect
to time).
5. Take the derivative of both sides of the equation with respect to
time. Each derivative of a variable will have a d_/dt.
6. Plug in all values for the given situation. In some cases, you will
need to do pythagorean theorem to find missing lengths. Pay close
attention to if values should be positive or negative. For example,
the rate of change of a decreasing area will be negative.
7. Solve for the missing value. Be sure to include units when
necessary.
Length = units
dL/dt = units/unit time
Ex. m/s
Area = units2
dA/dt = units2/unit time
Ex. m2/s
Volume = units3
dV/dt = units3/unit time
Ex. m3/s
Example 1
If 4x2y + 3x = -9 and dy/dt = -2 when x = 1 and y = -3, find dx/dt
Example 2:
Air is being pumped into a spherical balloon at a rate
of 5 cm3/min. Determine the rate at which the radius of the
balloon is increasing when the diameter of the balloon is 20 cm.
Example 3:
When a circular shield of bronze is heated over a fire its radius
increases at the rate of 1/5 cm/sec. At what rate is the shield's area
increasing when the radius is 50 cm?
Example 4
Example 5
Example 6
A rectangular trough is 10 ft long, 4 ft across the top, and 60 in deep.
If water flows in at a rate of 5 ft3/min, how fast is the depth rising
when the water is 2 ft deep?
Example 7
Example 8
A ladder 20 feet long leans against a building. If the bottom of the
ladder slides away from the building horizontally at a rate of 4 ft/sec,
how fast is the ladder sliding down the house when the top of the
ladder is 8 feet from the ground?