Chapter 5: Applications of the Derivative Derivatives Chapter 5: Applications

Objectives: To be able to use the derivative to analyze function Draw the graph of the function based on the analysis Apply the principles learned to problem situations

Example 1: The cross section of a water trough is an equilateral triangle with the top edge horizontal. If the trough is 10m long and 30 cm deep, and if water is flowing in at the rate of ¼ m3/min, how fast is the water level rising if the water in 20 cm at its deepest?

Example 2: How fast is the surface area of a cube is changing when the volume of the cube is 64 cu. Cm. and is increasing at the rate of 2 cc/sec?

Example 3: At 1 pm, Ship B is 25 km due north of ship A. If ship B is sailing west at the rate of 16 kph and ship A is sailing south at 20 kph, find the rate at which the distance between the 2 ships is changing at 1:30 pm.

Example 4: A ladder 5 m long leans against a vertical wall of a house. If the bottom of the ladder is pulled horizontally away from the house at 4 meters per sec, how fast is the ladder sliding when the bottom is 3 meters from the wall.

Example 5: a. how fast does his shadow lengthen? A man 6 ft tall walks away from a lamppost 15 ft high at the rate of 3 mph. a. how fast does his shadow lengthen? b. how fast does the tip of his shadow move?

Example 6: A water tank in the form of an inverted cone is being emptied at the rate of 6 m3/min. The altitude of the cone is 24 m and the radius is 12 m. Find how fast the water level is lowering when the water level is 10m deep.

Example 7: Boyle’s law for the expansion of gas is PV=C, where P is the number of pounds per square unit of pressure, V is the number of cubic units of the gas, and C is a constant. At a certain instant the pressure is 3000 lb/ft2, the volume is 5 ft3, and the volume is increasing at the rate of 3 ft3/min. Find the rate of change of the pressure at that instant.

Example 8: An automobile traveling at a rate of 30 ft/sec is approaching an intersection. When the automobile is 120 ft from the intersection, a truck traveling at the rate of 40 ft/sec crosses the intersection. The automobile and the truck are on roads that are at right angles to each other. How fast are the automobile and truck separating 2 sec after the truck leaves the intersection?

Example 9: A horizontal trough is 16 m long, and its ends are isosceles trapezoid with an altitude of 4 m, a lower base of 4 m, and an upper base of 6 m. If the water level is decreasing at the rate of 25 cm/min when the water is m deep, at what rate is the water being drawn from the trough?

Example 10: A man 6 ft tall is walking toward a building at the rate of 4 ft/sec. If there is a light on the ground 40 ft from the building, how fast is the man’s shadow on the building growing shorter when he is 30 ft from the building?

Example 11: An arc light hangs at a height of 30 ft above the center of the street 60 ft wide. A man, 6 ft tall, walks along the sidewalk at the rate of 4 ft/sec. How fast is his shadow lengthening when he is 40 ft up the street? How fast is the tip of his shadow moving?

Example 12: A particle P is moving along the graph of y = x2-4 where (x > 2) so that the x-coordinate of P is increasing at the rate of units per sec. How fast is the y-coordinate of P moving when x = 3?

Example 13: A long highway passes over a railroad track that is 100 ft below it and at right angles to it. A car traveling at 45 mph (66 ft per sec) is directly above a train moving at 60 mph (88 fps). How fast will they be separating after 10 seconds?

Example 14: Water is leaking out the bottom of a hemispherical tank of radius 8 ft at the rate of 2 cu ft per hr. The tank was full at a certain time. How fast is the water level changing when its height is 3 ft? Note: The volume of a spherical segment of height h is πh2(r-h/3).