Related Rates Section 3.9

When dealing with related rates… The procedure… Given information Unknown information Relationship between two quantities Derivative with respect to time Solve for unknown Substitute given information SOLVE!!! 

Example 1; Related Rates Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?

The radius of the balloon is increasing at the rate of .0127 cm/s. Example 1; Related Rates The radius of the balloon is increasing at the rate of .0127 cm/s.

Example 2; Related Rates 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/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall. dy dt 10ft y x dx dt

The ladder is going down the wall at the rate of ¾ ft/s. Example 2; Related Rates The ladder is going down the wall at the rate of ¾ ft/s.

Example 3; Related Rates A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m3/min, find the rate at which the water level is rising when the water is 3m deep. 2 4 r h

The water is rising at the rate of .28 m/min. Example 3; Related Rates The water is rising at the rate of .28 m/min.

Example 4; Related Rates Car A is traveling west at 50 mi/h and Car B is traveling north at 60 mi/h. Both are headed to the intersection of the roads. At what rate are the cars approaching each other when car A is .3mi and car B is .4 mi. from intersection. x=.3 mi Car A dx dt y=.4 mi dy dt Car B

Cars are nearing each other at a rate of 78 mi/h. Example 4; Related Rates Cars are nearing each other at a rate of 78 mi/h.

Example 5; Related Rates A man walks along a straight path at a speed of 4ft/s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight. x Man dx dt y θ dθ dt Light

The searchlight is rotating at a rate of .128 rad/s. Example 5; Related Rates 15 20 25 θ The searchlight is rotating at a rate of .128 rad/s.

3.9 Related Rates Summarize Notes Read section 3.9 Homework Pg.248 #1,3,5,7,9,11,13,15,17,23