§3.9 Related rates Main idea: In daily life, almost every quantity is a function of time t. If more than two quantities are related, how are their rates of change with respect to time t related? Our goal is to compute the rate of change of one quantity with respect to time using the rates of change of other quantities with respect to time.

Procedure for solving related rate problems Step 1: Get information. Introduce notation by drawing a figure, if appropriate, and assign variables to the quantities that vary. Be careful not to label a quantity with a number unless it never changes in the problem. Identify the quantities whose rates are given and the quantity whose rate to be found. Step 2: Find a relation that relates the variables. Eliminate unnecessary variables. Step 3: Differentiate both sides with respect to time. Step 4: Solve the rate to be found. Step 5: Do substitution.

Note: In step 3, we use CR since each quantity is a function of time t The rate of change of a quantity that is increasing is positive and the rate of change of a quantity that is decreasing is negative Only after differentiating and solving for required rate should we substitute the given values for those variables

Ex 1. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius of the ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area of the disturbed water changing? (Geometric shape: Circle) Solution:

Ex 2. A spherical snowball is melting in such a way that its volume is decreasing at a rate of 1 cm3/min. At what rate is the diameter changing when the diameter is 10 cm? (Geometric shape: Sphere) Solution: Step 1: Identify quantities that are related. One quantity is the volume of the snowball whose rate is given and another one is the diameter of the snowball whose rate we want. The figure shows all given information

Ex 3. A plane flying horizontally at an altitude of 6 miles will pass directly over a radar tracking station. If s is decreasing at a rate of 400 miles per hour when s is 10 miles, what is the speed of the plane? (Geometric shape: Triangle) Solution: Step 1: There are three quantities that are related in this example: altitude, distance between radar and plane, and the horizontal distance x. Caution: The altitude is constant and dx/dt = the velocity of the plane

Ex 4. A ladder 10 ft long rests against a vertical wall Ex 4. A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/sec. (1) How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? (See the textbook) (2) How fast is the angle between the ladder and the ground changing when the bottom of the ladder is 6 ft from the wall? (Geometric shape: Triangle)

Ex 5. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1m/s, how fast is the boat approaching the dock when it is 8 m from the dock? (Geometric shape: Triangle)

Ex 6. A paper cup has the shape of a cone with height 10 cm and radius 3 cm at the top. If water is poured into the cup at a rate of 2 cm3/s, how fast is the water level rising when the water is 5 cm deep? (Geometric shape: Cone) Solution:

Ex 7. A block of ice in the shape of a cube is melting so that its volume is decreasing at a rate of 25 cm3/sec. At what rate is its surface area decreasing when the edge of the cube is 10 cm? (Geometric shape: other cases) Solution: