Investigating Rates of Change
Related Rates:
Use your knowledge of derivatives and the indicated relationships
to investigate the following rates of change for the specified quantities.
1.
_____________________ An ice cube is melting at the rate of 0.3888 cm3
per minute. What is its instantaneous width when its side lengths are
shrinking at the rate of 0.01 cm per minute?
2.
______________________ A spherical balloon is being inflated at the
rate of 48 ft3 per minute. How fast is the surface area changing
when the instantaneous radius is 10 ft?
4
[Hints: Volumesphere = 3  r3
Surface Areasphere = 4  r2 ]
3.
______________________ A right cylindrical tank 24 feet in diameter
and filled with water is draining at the rate of 113.09734 cubic feet
per minute. How quickly is the water level falling?
[ Hint: Volumecylinder =  r2 h ]
4.
_______________________ A mechanical winch
is pulling a boat to the pier from a fixed
point 14 meters above the mooring to the
boat by drawing in the mooring line at a
steady rate of 6 meters per minute. How
fast is the boat approaching the pier when
the boat is 48 meters from the dock?
pulling rate = 6 m/min.
to winch
mooring line
? approach rate
14 m
Dock
2nd Base
5.
_______________________ A Pee Wee baseball diamond
is square in shape like a conventional baseball field
but the distance between consecutive bases is only 72
feet. If, when running from first to second base, Don
Junior reaches his top speed of 10 feet per second
when he is 54 feet from first base, then how fast is his
distance from home plate changing?
Home Plate
6.
________________ A man at the Marriot Inn descending from the 12th
floor in a glass elevator dropping at the rate of 5 feet per second
sees his family standing at the concierge’s desk 27 feet from the
base of the elevator. How fast is the line-of-sight distance between
him and his family changing when the elevator is 120 feet above
ground level?
Type II Portfolio Assignment
IB Mathematics SL
Investigating Rates of Change
Reg Noland, Instructor/Author
Page 1 of 4
AP Calculus AB
7.
_____________________ An industrial cylindrical hot water tank with a
5-foot diameter and height of nine feet is being filled with water by a
pipe at the rate of 9.18 gallons per minute. If the drain valve is shut
off, how fast is the water level rising?
[ Hints: 7.48 gallon  1 ft3 = 1728 in3 and Volume cylinder =  r 2 h ]
8.
________________ The Marbleous Concrete Company uses inverted conical sand
tanks (apex pointed down) to fill its cement trucks. If each tank has circular base
of 18 feet in diameter, a height of twelve feet, and a constant dispensing rate of
35.785 cubic feet of sand per minute, then how fast is the
height of the sand in the tank decreasing when the height of
the sand in the tank is nine feet. (Assume that the volume of
sand is dry and empties uniformly, maintaining a conical
shape within the tank).
[Hints: consider the ratio of the radius to the height to
1
establish the relationship among the dimensions; V =  r2 h.]
3
9.
_________________ A cannonball shot from a
cannon with an elevation of 64.2° and muzzle
velocity of 49 meters/second from a height of
22.725 meters travels along a parabolic path
toward its target 175 meters away. How fast
is the projectile approaching its target when
it is six seconds into its flight?
[Hint: h(t) = h0 + vy0 t – 4.9 t 2]
10.
_____________________ Indiana Jones is in a square room in a tomb that
measures 18 ft by 18 ft with a 15 ft ceiling. He accidentally trips a trap
mechanism that shuts all entrances to the room and makes the ceiling
begin to descend at the rate of 0.5 ft/min. How fast is the volume of air
in the room changing when the ceiling is seven feet high?
11.
______________________ A spherical snowball is melting at the rate
of 6 ml per minute. How fast is the radius changing when the
instantaneous diameter is 24 cm?
4
[Hints: Volumesphere = 3  r3
Surface Areasphere = 4  r2 ]
Type II Portfolio Assignment
IB Mathematics SL
Investigating Rates of Change
Reg Noland, Instructor/Author
Page 2 of 4
AP Calculus AB
12.
______________________ A right conical tank 24 feet in diameter
and 16 feet tall was filled with water and is now draining
at the ridiculously accurate rate of 9.424777961 cubic
feet per minute. How quickly is the water level falling
when the water level is 4 ft. in height?
1
[ Hint: Volumecone = 3  r2 h ]
13.
_______________________ A mechanical winch
is pulling a boat to the pier from a fixed
point 35 meters above the mooring to the
boat by drawing in the mooring line at a
steady rate of 3 meters per minute. How
fast is the boat approaching the pier when
the boat is 120 meters from the dock?
14.
pulling rate = 3 m/min.
to winch
mooring line
35 m
? approach rate
Dock

____________________ When a plane passed over a
de
plane
radar tracking station, its altitude was measured
to be 35000 ft. As it flew down range, it remained
level at the constant altitude of 35000 ft. When it
passed over a check point 12000 feet down range,
it was moving away from the radar station at the
rate of 271.135 ft / sec.
What was its
instantaneous speed at that point?
35000 ft
12000 ft
Radar
Station
15.
16.
_______________________ A construction worker on
the roof of a building is using a mechanical winch to
pull a 50 foot ladder up the side of the building. If
the winch is pulling at the rate of 8 feet per minute,
how fast is the foot of the ladder sliding along the
ground toward the building when the head of the
ladder is 14 feet up the wall from the ground?
Worker with winch
Rope
50 ft Ladder
14 ft.
_____________________ A radar station begins tracking a small plane
approaching from the west at an altitude of 4 miles and traveling
at a ground speed of 240 mph. How fast is the angle of elevation
changing when the angle  is 60°?
Type II Portfolio Assignment
IB Mathematics SL
Investigating Rates of Change
Reg Noland, Instructor/Author
Check
Point


Page 3 of 4
AP Calculus AB
17.
_________________ A hot-air balloon is descending straight down to
a landing pad 80 feet from an observer. If the observer sees the balloon
descend at a rate of 3.6° per minute (remember to convert to radians),
to the nearest tenth, how fast is the balloon descending when it is 150
ft. above ground level?
18.
_________________ A police car beacon revolves at the rate of 12 revolutions
per minute. If the police car is parked 80 feet from a wall, how fast is the light
beam moving along the wall as it strikes a point 39 ft in front of the police
car?
19.
_________________ Huck Finn throws a pebble into a still pond
creating ripples that form concentric circles from the point of
impact. If the radius of the lead (initial) ripple is increasing at
the rate of 6 feet per second, how fast is the disturbed area of
the pond increasing after 1.5 seconds?
20.
_________________ The winch that reels up an elevator cable has a diameter
of 3 feet and rotates at the speed of 17 revolutions per minute. How fast
does the elevator descend in the shaft? (Express your answer in feet per
second to the nearest hundredth.)
21.
____________________ A Southwest Airlines jumbo jet airbus is traveling in still air
(ha! ha!) with an air speed of 360 mph when it begins its gradual descent at a 7°
angle. How fast is it losing altitude?

 = 7°
360 mph
22.
_________________ A radar station begins tracking a small plane 
3
approaching from the west when it is 27 11 miles down range at
an altitude of 17000 ft. and traveling at 167 mph. How fast must
the radar dish rotate in order to continue tracking the plane?
Type II Portfolio Assignment
IB Mathematics SL
Investigating Rates of Change
Reg Noland, Instructor/Author
Page 4 of 4
AP Calculus AB