3. For a certain rectangle the length of one side is always three
times the length of the other side.
(a) If
the shorter side is decreasing at a rate of 2 inches/minute at what rate is the
longer side decreasing?
(b) At
what rate is the enclosed area decreasing when the shorter side is 6 inches
long and is decreasing at a rate of 2 inches/minute?
[Solution]
4. A thin sheet of ice is in the form of a circle. If the ice is melting in such a way that the
area of the sheet is decreasing at a rate of 0.5 m^{2}/sec at what rate
is the radius decreasing when the area of the sheet is 12 m^{2 }? [Solution]
5. A person is standing 350 feet away from a model rocket
that is fired straight up into the air at a rate of 15 ft/sec. At what rate is the distance between the
person and the rocket increasing (a)
20 seconds after liftoff? (b) 1
minute after liftoff? [Solution]
6. A plane is 750 meters in the air flying parallel to the
ground at a speed of 100 m/s and is initially 2.5 kilometers away from a radar
station. At what rate is the distance
between the plane and the radar station changing (a) initially and (b) 30
seconds after it passes over the radar station?
See the (probably bad) sketch below to help visualize the problem. [Solution]
7. Two people are at an elevator. At the same time one person starts to walk
away from the elevator at a rate of 2 ft/sec and the other person starts going
up in the elevator at a rate of 7 ft/sec.
What rate is the distance between the two people changing 15 seconds
later? [Solution]
8. Two people on bikes are at the same place. One of the bikers starts riding directly
north at a rate of 8 m/sec. Five seconds
after the first biker started riding north the second starts to ride directly
east at a rate of 5 m/sec. At what rate
is the distance between the two riders increasing 20 seconds after the second
person started riding? [Solution]
9. A light is mounted on a wall 5 meters above the
ground. A 2 meter tall person is
initially 10 meters from the wall and is moving towards the wall at a rate of
0.5 m/sec. After 4 seconds of moving is
the tip of the shadow moving (a)
towards or away from the person and (b)
towards or away from the wall? [Solution]
10. A tank of water in the shape of a cone is being filled
with water at a rate of 12 m^{3}/sec.
The base radius of the tank is 26 meters and the height of the tank is 8
meters. At what rate is the depth of the
water in the tank changing when the radius of the top of the water is 10
meters? [Solution]
11. The angle of elevation is the angle formed by a
horizontal line and a line joining the observer’s eye to an object above the
horizontal line. A person is 500 feet
way from the launch point of a hot air balloon.
The hot air balloon is starting to come back down at a rate of 15
ft/sec. At what rate is the angle of
elevation, ,
changing when the hot air balloon is 200 feet above the ground. See the (probably bad) sketch below to help
visualize the angle of elevation if you are having trouble seeing it. [Solution]