Related+Rates

= Related Rates =

**Strategy for Solving Related Rate Problems**

 * Understand the problem
 * In particular, identify the variable whose rate of change you seek and the variable whose rate of change you know.
 * Develop a mathematical model of the problem.
 * Draw a picture and label the parts that are important to the problem. Be sure to distinguish constant quantities from variables that change over time. Only constant quantities can be assigned numerical values at the start.
 * Write an equation relating the variable whose rate of change you seek with the variable(s) whose rate of change you know.
 * The formula is often geometric but it could come from a scientific application.
 * Differentiate both sides of the equation implicitly with respect to time t.
 * Be sure to follow all the differentiation rules. The Chain Rule will be especially critical, as you will be differentiating with respect to the parameter t.
 * Substitute values for any quantities that depend on time.
 * Notice that it is only safe to do this after the differentiation step. Substituting too soon "freezes the picture" and makes changeable variables behave like constants, with zero derivatives.
 * Interpret the solution
 * Translate your mathematical result into the problem setting and decide whether the result makes sense.

Example Problem
A 13-foot ladder is leaning against a house when its base starts to slide away. By the time the base is 12 feet from the house, the base is moving at a rate of 5 ft/sec. How fast is the top of the ladder sliding down the wall at that moment? dx/dt=5 (Given) x=12 dz/dt=0 (Length of the ladder is not changing) z=13 dy/dt=? (This is what we are looking for) y=? (Must find the length of y at the time during which the base of the ladder is 12 feet from the house.)


 * We are looking for a rate of change of the length of a side of the triangle so we need to use an equation involving the sides of the triangle to solve the problem.
 * a^2+b^2=c^2
 * Substitute what we know to find the length of the third side when the base of the ladder is 12 feet from the house.
 * 12^2+y^2=13^2
 * 144+y^2=169
 * y^2=25
 * y=5
 * Implicity differentiate this equation.
 * 2x(dx/dt)+2y(-dy/dt)=2z(dz/dt)
 * Substitute what we know and solve for the missing rate.
 * 2(12)(-5)+2(5)(dy/dt)=2(13)(0)
 * 120+10(dy/dt)=0
 * 10dy/dt=-120
 * dy/dt=12

Works Cited Finney, R.L., Demana, F.D., Waits, B.K., & Kennedy, D. (2007). //Calculus: graphical, numerical, algebraic//. Boston, Massachusetts: Pearson Prentice Hall.