Related Rates

Steps to use in solving related rates problems.

  1. Draw and label a figure.
  2. Write what the problem states as given in mathematical terms.  Also, write what you are trying to find in mathematical terms.
  3. State an equation that is valid at any time for the variables in the problem.
  4. Differentiate with respect to the appropriate variable.
  5. Substitute in values given in the problem.
  6. Solve the resulting equation for the variable you are asked to find.

Example
If one leg of a right triangle increases at the rate of 2 in/sec, while the other leg decreases at 3 in/sec, find how fast the hypotenuse is changing when the first leg is 6 ft and the other leg is 8 ft.

  1. Draw a right triangle and label the legs u and v.  Label the hypotenuse z.
  2. du/dt = 1/6 ft/sec
    dv/dt = -1/4 ft/sec
    Find dz/dt
  3. At any time,   u2 + v2 = z2
    Find z = 10 by substituting the appropriate values in the equation in #3.
  4. 2z dz/dt = 2u du/dt + 2v dv/dt
  5. 2(10) dz/dt = 2(6) (1/6) + 2(8)(-1/4)
        10 dz/dt = 1 -2
             dz/dt = -1/10 ft/sec

Problems