Warm-Up If x 2 + y 2 = 25, what is the value of d 2 y at the point (4,3)? dx 2 a) -25/27 c) 7/27 e) 25/27 b) -7/27 d) 3/4

Warm-Up If x 2 + y 2 = 25, what is the value of d 2 y at the point (4,3)? dx 2 a) -25/27 c) 7/27 e) 25/27 b) -7/27 d) 3/4

2-6: Related Rates Objectives: Find related rates Solve problems using related rates ©2002 Roy L. Gover

Definition Two or more variables that vary with each other and vary with time are related rates of change.

Example The rate of change of the volume of water in the cone is related to the rates of change of the radius h r and the height of the water.

Example The rate of change of the angle of elevation of a camera photographing the ascent of a hot air balloon is related to the rate of change of the balloon’s height

Important Idea A rate of change of a variable with respect to time is the derivative of the variable with respect to time written as Examples:

Example Assume that x and y are differentiable functions of t. Find the indicated values of and for 1.Find when x =3 given =2

Example Assume that x and y are differentiable functions of t. Find the indicated values of and for 2.Find when x =1 given =5

Example Find the indicated values of and for 2.Find when x =1 given =5

Important Idea You make the substitutions after you find the derivative.

Try This Assume that x and y are differentiable functions of t. Find the indicated values of and for 1.Find when x =3 & y =4 given =8

Try This Assume that x and y are differentiable functions of t. Find the indicated values of and for 2.Find when x =4 & y =3 given =2

Example A pebble is dropped in a pond causing ripples that expand in the form of circles. The radius, r, of the outer ripple is increasing at a constant rate of 2 ft./sec. At what rate is the area of the disturbed water increasing when the radius is 6 feet?

Steps for Solving Related Rate Problems 1. Label all given rates and rates to be found as. Make a sketch. These steps must be performed in the order given.

Steps for Solving Related Rate Problems 2. Write an equation that shows a relationship between the variables.

Steps for Solving Related Rate Problems 3. Implicitly differentiate with respect to time both sides of the relating equation using the chain rule.

Steps for Solving Related Rate Problems 4. Substitute all known values given for variables and rates of change after differentiating. Sometimes you may need to solve for the needed variables.

Steps for Solving Related Rate Problems 5. Solve, if necessary, for the required rate of change. Use the correct unit of measure.

Example Air is being released from a spherical balloon at the rate of 3 cu. in./min. What is the rate of change of the radius of the balloon when the radius of the balloon is 2 in.?

Example Air is being released from a spherical balloon at the rate of 3 cu. in./min. What is the rate of change of the radius of the balloon when the volume of air in the balloon is 36  cu. in.?

Example An airplane is flying toward a radar station at an altitude of 2 mi. If the distance, s, from the plane to the radar is decreasing at a rate of 300 mph, what is the speed of the plane when s is 3 mi. ?

Try This An airplane is flying away from a radar station at an altitude of 2 mi. If the distance, s, from the plane to the radar is increasing at a rate of 500 mph, what is the speed of the plane when s is 10 mi. ?

Solution s x 2 mi.

Example A TV camera 2000 ft. from the launchpad is filming the lift-off of the shuttle. The shuttle is rising vertically according to the equation. Find the rate of change of the camera’s angle of elevation 10 sec. after liftoff.

Example A trough is 12 ft. long and 3 ft. across the top. Its ends are isosceles triangles with altitude of 3 ft. Water is being pumped into the trough at 2 cu. ft./min. How fast is the water level rising when it is 1 ft. deep? …

Example 12’ 3’ h w Relating Equa.:

Example 12’ 3’ h w By similar triangles:

Try This h r R H A water tank in the form of a right circular cone with radius, R, of 6’ and height, H, of 18’ is leaking water at the rate of 2 cu.ft./hr. What is the rate of change of h when r =4’?

Solution

Lesson Close Name some things that are important to remember when doing related rate problems.

Assignment