Steps to Solving Related Rates Problems
               Example: Aladder, 10 feet long, is leaning against a building. The bottom of the ladder is moved
               away from the building at the constant rate of 1 ft/sec. Find the rate at which the ladder is falling
                                                             2
               down the building when the ladder is 6 feet from the building.
                Step 1:                                                   Ex:
                Drawapicture of the situation, and label in con-
                junction with Step 2.
                Step 2:                                                   Ex:
                Write down all known variables (things that can           x=“distance from ladder to wall” (x = 6)
                change) and quantities (things that stay con-             dx = “rate ladder is being pulled” (dx = 1)
                                                                           dt                         √       dt   2
                stant), including any given rates of change.                                              2    2
                                                                          h=“height of ladder” (h =     10 −6 =8)
                                                                          dh = “rate ladder is falling”
                                                                           dt
                                                                          10 ft = “length of ladder”
                Step 3:                                                   Ex:
                Identify what the problem is asking you to find.           Weareseeking the rate at which the ladder falls,
                                                                          so dh = ?
                                                                             dt
                Step 4:                                                   Ex:
                                                                           2    2      2
                Write the “main” equation relating the necessary          x +h =10
                variables.
                Step 5:                                                   Ex:
                (Only used sometimes) Use a second equation to            Not used in this example.
                write the main equation in terms of one variable.
                Step 6:                                                   Ex:
                                                                           d  2    2    d
                Differentiate with respect to the independent              dt[x +h ] = dt[100]
                variable (usually time t).                                2xdx +2hdh = 0
                                                                             dt     dt
                Step 7:                                                   Ex:
                Substitute given values into derived equation and         2(6)(1)+2(8)dh = 0
                                                                               2        dt
                solve for the desired value.                              16dh = −6 =⇒ dh =−3
                                                                             dt            dt     8
                                                                          (The negative answer indicates that h is
                                                                          decreasing, so the ladder is falling.)
                Step 8:                                                   Ex:
                Answer the question as stated.                            The ladder is falling at a rate of 3 ft/sec.
                                                                                                          8
                                                                          (We omit the negative since we specify in our
                                                                          answer that the ladder is falling.)
                        Don’t forget to differentiate before you substitute in the values of the variables!