Chapter 5:
Applications of the
Derivative
Chapter 4:
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 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).