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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!