1 Related Rates Section 2.6

2 Related Rates (Preliminary Notes) If y depends on time t, then its derivative, dy/dt, is called a time rate of change. Usually, related rate problems will require implicit differentiation. Remember, dy/dt can be read as the change in y with respect to time.

3 Common Formulas Used in Related Rate Problems Perimeter of rectangle: Area of circle: Volume of cube: Volume of cylinder: Volume of cone: Volume of sphere: Pythagorean theorem:

4 Strategy For Solving Related Rate Problems Draw a picture Write down the given quantities and identify the quantity to find Relate the variables by writing an equation Relate the rates by implicitly differentiating with respect to time Solve for the unknown quantity Substitute in the known values to find the unknown Tag on the appropriate units

5 Expanding Rectangle… Example 1: The length of a rectangle is increasing at a rate of 3ft/sec and the width is increasing at 2ft/sec. When the length is 4 ft and the width is 3ft, how fast are the perimeter, area and the length of a diagonal increasing? x = length y = width Given: Step 1: Draw a picture Step 2: Write down the givens and the quantity to find

6 Perimeter… a)Perimeter… x = 4 y = 3 Implicitly differentiate with respect to time Find:The rate the perimeter is increasing when x = 4 and y = 3. Step 2 Continued: …write down the quantity to find Step 3: Relate the variables Step 4: Relate the rates Write an equation

7 Perimeter… x=4 y=3 Step 5: Solve for the unknown Already solved in this example Step 6: Substitute into rate equation Step 7: Tag on units

8 Area… x = 4 y = 3 x = 4 y = 3 x = length y = width Given: b)Area… Find: The rate the area is increasing when x = 4 and y = 3. Step 2 Continued: …write down the quantity to find Step 3: Relate the variables Step 4: Relate the rates Step 5: Solve for the unknown Already solved in this example Step 6: Substitute into rate equation Step 7: Tag on units

9 Diagonal Length… x yD x = 4 y = 3 Step 3: Relate the variables Step 4: Relate the rates x = 4 y = 3 D = Step 2 (cont) Find: The rate the diagonal is increasing when x = 4 and y = 3. Step 5: Solve for the unknown Step 6: Substitute into rate equation

10 Increasing the Area of a Circle Example 2 The radius of a circle is increasing at the rate of 5 inches per minute. At what rate is the area increasing when the radius is 10 inches? 1) Given: 2) Find: 3) Relate the variables (formula): 4) Relate the rates (implicit differentiation): 5) Substitute into rate equation:

11 Blowing Up a Balloon Example 3: Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet. R R= 2 3) Relate the variables (formula) 4) Relate the rates (implicit differentiation) R= 2 1) Given: 2) Find: 5) Solve for the unknown 6) Substitute into rate equation:

12 The Sliding Ladder…part a) 5 x y Example 4: A 5 meter-long ladder is leaning against the side of a house. The foot of the ladder is pulled away from the house at a rate of 0.4 m/s. a) Determine how fast the top of the ladder is descending when the foot of the ladder is 3 meters from the house. x = 3 x = 3 y = 1) Given: 2) Find: 3) Relate the variables (formula) 4) Relate the rates (implicit differentiation) 5) Solve for the unknown 6) Substitute into rate equation:

13 The Sliding Ladder…part b) x y Example 4: b) Determine the rate at which the angle between the ladder and the wall is changing when the base of the ladder is 3 m from the house. x = 3 x = 3 y =4 1) Given: 2) Find: 3) Relate the variables (formula) 4) Relate the rates (implicit differentiation) 5) Solve for the unknown 6) Substitute into rate equation: 5

14 Volume of a Conical Sand Pile Example 5: Sand is falling off a conveyor onto a conical pile at the rate of 15 cubic feet per minute. The diameter of the base of the cone is approximately twice the altitude. At what rate is the height of the pile changing when it is 10 feet high? 2r h 1) Given: 2) Find: h = 10

15 Homework Section 2.6 page 149 #3, 5, 15, 19, 20, 23, and 27