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Observe that the rate of change is a maximum when cos (t) is a maximum. Since
2
the maximum of the cosine function is 1, the maximum value of cos (t)is1,
occurring when = 0. We conclude that the maximum rate of angle change is 1.32
radians/second. This occurs when = 0, that is, when the jet reaches its closest
point to the observer. (Think about this; it should match your intuition!) Since
humans can track objects at up to about 3 radians/second, this means that we can
visually follow even a fast jet at a very small distance.

EXERCISES 4.8

7. A 10 ft ladder leans against the side of a building as in ex-
WRITING EXERCISES
ample 8.2. If the bottom of the ladder is pulled away from
1. As you read examples 8.1–8.3, to what extent do you find the wall at the rate of 3 ft/sec and the ladder remains in con-
the pictures helpful? In particular, would it be clear what x tact with the wall, (a) find the rate at which the top of the
and y represent in example 8.3 without a sketch? Also, in ladder is dropping when the bottom is 6 ft from the wall.
′
′
′
example 8.3 explain why the derivatives x (t), y (t) and d (t) (b) Find the rate at which the angle between the ladder and
are all negative. Does the sketch help in this explanation? the horizontal is changing when the bottom of the ladder is
6 ft from the wall.
2. In example 8.4, the increase in advertising dollars from year
1 to year 2 was AED 1500. Explain why this amount is not 8. Two buildings of height 20 ft and 40 ft, respectively, are
′
especially relevant to the approximation of s (4). 60 ft apart. Suppose that the intensity of light at a point
between the buildings is proportional to the angle in the
figure. (a) If a person is moving from right to left at 4 ft/s, at
1. Oil spills out of a tanker at the rate of 120 gl/min per what rate is changing when the person is exactly halfway
minute. The oil spreads in a circle with a thickness of 1 ′′ . between the two buildings? (b) Find the location at which
4
3
Given that 1 ft equals 7.5 gallons, determine the rate at the angle is maximum.
which the radius of the spill is increasing when the radius
reaches (a) 100 ft and (b) 200 ft. Explain why the rate de-
creases as the radius increases.
2. Oil spills out of a tanker at the rate of 90 gallon per minute. 40'
The oil spreads in a circle with a thickness of 1 ′′ . Determine 20' θ
8
the rate at which the radius of the spill is increasing when 60'
the radius reaches 100 feet.
3. Oil spills out of a tanker at the rate of g gallons per minute. 9. A plane is located x = 40 mile (horizontally) away from an
The oil spreads in a circle with a thickness of 1 ′′ . (a) Given airport at an altitude of h mile. Radar at the airport de-
4
that the radius of the spill is increasing at a rate of 0.6 ft/min tects that the distance s(t) between the plane and airport
when the radius equals 100 feet, determine the value of g. is changing at the rate of s (t) =−240 mph. (a) If the plane
′
(b) If the thickness of the oil is doubled, how does the rate flies toward the airport at the constant altitude h = 4, what
of increase of the radius change? is the speed |x (t)| of the airplane? (b) Repeat with a height
′
4. Assume that the infected area of an injury is circular. (a) If of 6 mile. Based on your answers, how important is it to
the radius of the infected area is 3 mm and growing at a know the actual height of the airplane?
rate of 1 mm/hr, at what rate is the infected area increas- 10. (a) Rework example 8.3 if the police car is not moving. Does
ing? (b) Find the rate of increase of the infected area when this make the radar gun’s measurement more accurate?
the radius reaches 6 mm. Explain in commonsense terms (b) Show that the radar gun of example 8.3 gives the cor-
why this rate is larger than that of part (a). rect speed if the police car is located at the origin.
5. Suppose that a raindrop evaporates in such a way that it 11. Show that the radar gun of example 8.3 gives the correct
maintains a spherical shape. Given that the volume of a speed if the police car is at x = 1 moving at a speed of
4
sphere of radius r is V = r and its surface area is A = ( 2 − 1) 50 mph. 2
3
√
3
′
2
′
4 r , if the radius changes in time, show that V = Ar . If 12. Find a position and speed for which the radar gun of exam-
′
the rate of evaporation (V ) is proportional to the surface
Copyright © McGraw-Hill Education Copyright © McGraw-Hill Education 6. Suppose a forest fire spreads in a circle with radius 13. For a small company spending AED x thousand per year in
ple 8.3 has a slower reading than the actual speed.
area, show that the radius changes at a constant rate.
advertising, suppose that annual sales in thousands of dol-
changing at a rate of 5 ft/min. When the radius reaches
−0.05x
. The three most recent yearly
lars equal s = 60 − 40e
200 feet, at what rate is the area of the burning region
advertising figures are given in the table.
increasing?
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