65
Chapter 8: Applications of Derivatives
439. At noon, Ship A is 150 kilometers east of
Ship B. Ship A is sailing west at 20 kilometers per hour, and Ship B is sailing north at
35 kilometers per hour. How quickly is the
distance between them changing at 3 p.m.?
Round your answer to the nearest
hundredth.
440. A particle moves along the curve y x
=
+
3
1.
As the particle passes through the point
(8, 3), the x coordinate is increasing at a rate
of 5 centimeters per second. How quickly is
the distance from the particle to the origin
changing at this point? Round your answer
to the nearest hundredth.
441. Two people start walking from the same
point. One person walks west at 2 miles per
hour, and the other walks southwest (at an
angle 45° south of west) at 4 miles per hour.
How quickly is the distance between them
changing after 40 minutes? Round your
answer to the nearest hundredth.
442. A trough is 20 feet long, and its ends are
isosceles triangles that are 5 feet across the
top and have a height of 2 feet. If the trough
is being filled with water at a rate of 8 cubic
feet per minute, how quickly is the water
level rising when the water is 1 foot deep?
434. If y = x
4
+ 3x
2
+ x and dx
dt
= 4, find
dy
dt
when
x = 3.
435. If z
3
= x
2
– y
2
, dx
dt
= 3, and
dy
dt
= 2, find dz
dt
when x = 4 and y = 1.
436. Two sides of a triangle are 6 meters and
8 meters in length, and the angle between
them is increasing at a rate of 0.12 radians
per second. Find the rate at which the area
of the triangle is increasing when the angle
between the sides is π
6
. Round your answer
to the nearest hundredth.
437. A ladder 8 feet long rests against a vertical
wall. If the bottom of the ladder slides away
from the wall at a rate of 3 feet per second,
how fast is the angle between the top of the
ladder and the wall changing when the
angle is π
3
radians?
438. The base of a triangle is increasing at a rate
of 2 centimeters per minute, and the height
is increasing at a rate of 4 centimeters per
minute. At what rate is the area changing
when b = 20 centimeters and h = 32
centimeters?
Chapter 8: Applications of Derivatives
439. At noon, Ship A is 150 kilometers east of
Ship B. Ship A is sailing west at 20 kilometers per hour, and Ship B is sailing north at
35 kilometers per hour. How quickly is the
distance between them changing at 3 p.m.?
Round your answer to the nearest
hundredth.
440. A particle moves along the curve y x
=
+
3
1.
As the particle passes through the point
(8, 3), the x coordinate is increasing at a rate
of 5 centimeters per second. How quickly is
the distance from the particle to the origin
changing at this point? Round your answer
to the nearest hundredth.
441. Two people start walking from the same
point. One person walks west at 2 miles per
hour, and the other walks southwest (at an
angle 45° south of west) at 4 miles per hour.
How quickly is the distance between them
changing after 40 minutes? Round your
answer to the nearest hundredth.
442. A trough is 20 feet long, and its ends are
isosceles triangles that are 5 feet across the
top and have a height of 2 feet. If the trough
is being filled with water at a rate of 8 cubic
feet per minute, how quickly is the water
level rising when the water is 1 foot deep?
434. If y = x
4
+ 3x
2
+ x and dx
dt
= 4, find
dy
dt
when
x = 3.
435. If z
3
= x
2
– y
2
, dx
dt
= 3, and
dy
dt
= 2, find dz
dt
when x = 4 and y = 1.
436. Two sides of a triangle are 6 meters and
8 meters in length, and the angle between
them is increasing at a rate of 0.12 radians
per second. Find the rate at which the area
of the triangle is increasing when the angle
between the sides is π
6
. Round your answer
to the nearest hundredth.
437. A ladder 8 feet long rests against a vertical
wall. If the bottom of the ladder slides away
from the wall at a rate of 3 feet per second,
how fast is the angle between the top of the
ladder and the wall changing when the
angle is π
3
radians?
438. The base of a triangle is increasing at a rate
of 2 centimeters per minute, and the height
is increasing at a rate of 4 centimeters per
minute. At what rate is the area changing
when b = 20 centimeters and h = 32
centimeters?