1. 5.5 Section 4.6 – Related Rates
I can use implicit differentiation to solve related rate word problems.
Day 1:
Find the slope the following at x = 1:
5.5

6. The ladder is moving away at a rate of
A 14 foot ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2 ft/s, how fast will the end be moving away from the wall when the top is 6 ft above the ground? x y L 14 6
The ladder is moving away at a rate of

7. A man 6 ft tall is walking at a rate of 2 ft/s toward a street light 16 ft tall. At what rate is the size of his shadow changing? x y 16 6
The size of his shadow is reducing at a rate of 6/5.

8. A boat whose deck is 10 ft below the level of a dock, is being drawn in by means of a rope attached to a pulley on the dock. When the boat is 24 ft away and approaching the dock at ½ ft/sec, how fast is the rope being pulled in? 24 x y R
-10 26
The rope is being pulled in at a rate of 6/13

9. A pebble is dropped into a still pool and sends out a circular ripple whose radius increases at a constant rate of 4 ft/s. How fast is the area of the region enclosed by the ripple increasing at the end of 8 seconds.
At t = 8, r = (8)(4) = 32
The area is increasing at a rate of

10. A spherical container is deflated such that its radius decreases at a constant rate of 10 cm/min. At what rate must air be removed when the radius is 5 cm? 5
Air must be removed at a rate of

11. A ruptured pipe of an offshore oil platform spills oil in a circular pattern whose radius increases at a constant rate of 4 ft/sec. How fast is the area of the spill increasing when the radius of the spill is 100 ft?
The area of the spill is increasing at a rate of

12. Sand pours into a conical pile whose height is always one half its diameter. If the height increases at a constant rate of 4 ft/min, at what rate is sand pouring from the chute when the pile is 15 ft high? 15 15
The sand is pouring from the chute at a rate of

13. The depth of the liquid is decreasing at a rate of
Liquid is pouring through a cone shaped filter at a rate of 3 cubic inches per minute. Assume that the height of the cone is 12 inches and the radius of the base of the cone is 3 inches. How rapidly is the depth of the liquid in the filter decreasing when the level is 6 inches deep? 3 r 12 h
The depth of the liquid is decreasing at a rate of

14. 6

15. If and x is decreasing at the rate of 3 units per second,
the rate at which y is changing when y = 2 is nearest to:
a. –0.6 u/s b. –0.2 u/s c u/s d u/s e u/s

16. When a wholesale producer market has x crates of lettuce available on a given day, it charges p dollars per crate as determined by the supply equation
If the daily supply is decreasing at the rate of 8 crates per day, at what rate is the price changing when the supply is 100 crates?