Section 2-6 Related Rates

Related Rates Definition: When two or more related variables are changing with respect to time they are called related rates In related rate story problems, the idea is to find a rate of change (with respect to time) of one quantity by using the rate of change (with respect to time) of a related quantity.

Derivative = rate of change When we say “rate of change”, we mean “rate of change with respect to time”. example: velocity- change in distance with respect to time If the variable is different from time use implicit differentiation

Helpful suggestions for solving Related Rates Identify all known and unknown quantities. Make a sketch and label it. Find an equation that describes the relationship between the variables. Common equations include the Pythagorean Theorem, Area of circle, Volumes of known geometric shape, SOH-CAH-TOA, similar triangles, etc. Differentiate both sides of the equation with respect to time. Substitute all known quantities and rates into the equation and solve.

1) Differentiate each with respect to t.

 

 

4) A 13 – foot ladder is leaning against the wall of a house 4) A 13 – foot ladder is leaning against the wall of a house. The base of the ladder slides away from the wall at a rate of 0.75 feet per second. How fast is the top of the ladder moving down the wall when the base is 12 feet from the wall?

5) Oil spills into a lake in a circular pattern 5) Oil spills into a lake in a circular pattern. If the radius of the circle increases at a constant rate of 3 feet per second, how fast is the area of the spill increasing at the end of 30 minutes.

6) A baseball diamond is a square with side 90 feet 6) A baseball diamond is a square with side 90 feet. A batter hits the ball and runs toward first base with a speed of 24 feet per second. At what rate is his distance from second base decreasing when he is halfway to first base?

7) A weather balloon is released 50 feet from an observer 7) A weather balloon is released 50 feet from an observer. It rises at a rate of 8 feet per second. How fast is the angle of elevation changing when the balloon is 50 feet high? Ɵ

8) Two cars start moving from the same point at noon 8) Two cars start moving from the same point at noon. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing at 2:00 pm?

9) Air is being pumped into a spherical ball at a rate of 5 cubic centimeters per minute. Determine the rate at which the radius of the ball is increasing when the diameter of the ball is 20 cm.

10) Water is being poured into a conical reservoir at the rate of pi cubic feet per second.  The reservoir has a radius of 6 feet across the top and a height of 12 feet.  At what rate is the depth of the water increasing when the depth is 6 feet?

Assignment Page 154 # 1, 3,13,17, 20, 29, 31, and 43 and Assignment 2-6 on Related Rates