Compute the rate of change of one quantity in terms of the rate of change of another quantity.

Transcription

1 3.10 Related Rates Compute the rate of change of one quantity in terms of the rate of change of another quantity. Example 1: If x 2 y x + 4 = 0 and dx/dt = 3, find dy/dt when x = 1. Example 2: Air is being pumped into a balloon (we ll assume that it is spherical) so that its surface area increases at a rate of 2 cm 2 /s. At what rate is the radius of the balloon changing when its diameter is 10 cm? Example 3: A meter stick 3 ft long rests against a vertical wall. If the bottom of the stick slides away from the wall at a rate of ½ ft/s, determine the speed at which the top of the stick sliding down the wall when the top of the stick is (a) 1 ft from the floor? (b) ft from the floor?

2 Strategy 1. Read the problem carefully. 2. If possible, draw a diagram. 3. Assign notation. Anything that changes with time should be labeled with a symbol (i.e. A, V, r, x, y, z, etc.) 4. Express the required rate and the given information in terms of derivatives. 5. Write down an equation that relates the quantities together. It may be necessary to eliminate one of the variables in this step. 6. Differentiate both sides of the equation with respect to t using the Chain Rule. 7. Substitute the values from Step 4 into the resulting equation and solve for the unknown rate. 8. Check if your answer seems reasonable. Example 4: A water tank has the shape of an inverted circular cone with base radius 3 ft and height 9 ft. If water is being drained from the tank at a rate of 2 ft 3 /min, find the rate at which the water level is changing when the water is 6 ft deep. Example 5: A ground-based camera, positioned 40 m away, observes a space vehicle rise from the surface. When the vehicle is 80 m high, it is rises vertically at a speed of 10 m/s. (a) At what rate is the distance between the camera and the vehicle changing at that moment? (b) How fast must the camera angle of elevation change at this instant to keep the vehicle in sight?

3 Example 6: A particle is moving along the curve whose equation is 2 x 1 x e ln 3 y y e. While the x-coordinate of the particle is in the interval (0,3), it is increasing at a rate of 5 units/sec. At what rate is the y- coordinate of the particle changing when the particle is located at (1,1)? Example 7: The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at two o clock?

4 3.10 Related Rates Compute the rate of change of one quantity in terms of the rate of change of another quantity. Example 1: If x 2 y x + 4 = 0 and dx/dt = 3, find dy/dt when x = 1. Example 2: Air is being pumped into a balloon (we ll assume that it is spherical) so that its surface area increases at a rate of 2 cm 2 /s. How fast is the radius of the balloon increasing when its diameter is 10 cm?

5 Example 3: A meter stick 3 ft long rests against a vertical wall. If the bottom of the stick slides away from the wall at a rate of ½ ft/s, determine the speed at which the top of the stick sliding down the wall when the top of the stick is (a) 1 ft from the floor? (b) ft from the floor? Strategy 1. Read the problem carefully. 2. If possible, draw a diagram. 3. Assign notation. Anything that changes with time should be labeled with a symbol (i.e. A, V, r, x, y, z, etc.) 4. Express the required rate and the given information in terms of derivatives. 5. Write down an equation that relates the quantities together. It may be necessary to eliminate one of the variables in this step. 6. Differentiate both sides of the equation with respect to t using the Chain Rule. 7. Substitute the values from Step 4 into the resulting equation and solve for the unknown rate. 8. Check if your answer seems reasonable.

6 Example 4: A water tank has the shape of an inverted circular cone with base radius 3 ft and height 9 ft. If water is being drained from the tank at a rate of 2 ft 3 /min, find the rate at which the water level is changing when the water is 6 ft deep. Example 5: A ground-based camera, positioned 40 m away, observes a space vehicle rise from the surface. When the vehicle is 80 m high, it is rises vertically at a speed of 10 m/s. (a) At what rate is the distance between the camera and the vehicle changing at that moment? (b) How fast must the camera angle of elevation change at this instant to keep the vehicle in sight?

7 Example 6: A particle is moving along the curve whose equation is 2 x 1 x e ln 3 y y e. The graph of this curve is given to the right. While the x-coordinate of the particle is in the interval (0,3), it is increasing at a rate of 5 units/sec. At what rate is the y-coordinate of the particle changing when the particle is located at (1,1)? Example 7: The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at two o clock?