Related Rates Section 4.6a

Do Now Write the formula for distance from the origin to a point… and find its derivative! Suppose that the coordinates of the point (x and y) are both differentiable with respect to t. Then we can use the Chain Rule to find an equation that relates dD/dt, dx/dt, and dy/dt: Any equation involving two or more variables that are differentiable functions of time t can be used to find an equation that relates their corresponding rates!!!

Related Rates Problems Examples: How fast is a balloon rising at a given instant? How fast does the water level drop when a tank is drained at a certain rate? How fast does the surface area of a bubble increase when its volume increases at a certain rate? Questions like these require us to calculate a rate that may be difficult to directly measure, using a rate that we know, or is easy to measure  RELATED RATES!!!

Related Rates Problem Strategy 1. Draw a picture and name the variables and constants. Use t for time. Assume all variables are differentiable functions of t. 2. Write down the numerical information (in terms of the symbols you have chosen). 3. Write down what we are asked to find (usually a rate, expressed as a derivative).

Related Rates Problem Strategy 4. Write an equation that relates the variables. You may have to combine two or more equations to get a single equation that relates the variable whose rate you want to the variables whose rates you know. 5. Differentiate with respect to t. Then express the rate you want in terms of the rate and variables whose values you know. 6. Evaluate. Use known values to find the unknown rate.

Guided Practice When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm/sec. At what rate is the plate’s area increasing when the radius is 50 cm? Step 1: r = radius of plate, A = area of plate Step 2: At the instant in question, dr/dt = 0.01 cm/sec, r = 50 cm Step 3: Need dA/dt Step 4:

Guided Practice When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm/sec. At what rate is the plate’s area increasing when the radius is 50 cm? Step 5: Step 6: At the instant in question, the area is increasing at the rate of

Guided Practice A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the lift-off point. At the moment the range finder’s elevation angle is , the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment? rad/min when y when Range Finder 500 ft

Guided Practice A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the lift-off point. At the moment the range finder’s elevation angle is , the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment? y At the moment in question, the balloon is rising at the rate of 140 ft/min 500 ft

Guided Practice A police cruiser, approaching a right-angled intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between them and the car is increasing at 20mph. If the cruiser is moving at 60mph at the instant of measurement, what is the speed of the car? y Situation when x = 0.8, y = 0.6 s y x x

Guided Practice y x Situation when x = 0.8, y = 0.6 s y x x Pythagorean Theorem:

Guided Practice y x Situation when x = 0.8, y = 0.6 s y x x At the moment in question, the car’s speed is 70 mph

Guided Practice Water runs into a conical tank at the rate of 9 . The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep? 5 ft x 10 ft y when y = 6 ft

Guided Practice Water runs into a conical tank at the rate of 9 . The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep? 5 ft x Similar Triangles: 10 ft y

Guided Practice Water runs into a conical tank at the rate of 9 . The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep? 5 ft x 10 ft y At this moment, the water level is rising at about 0.318 ft/min.