AP Calculus AB
Unit 5 – More Applications of Derivatives
Assignment 5.1
1.) If a box with a square base and an open top is to have a volume of 4 ft. 3, find the dimensions that require
the least material. (Disregard the thickness of the material and waste in construction.)
2.) One thousand feet of chain link fence will be used to construct six cages for a zoo exhibit as shown in the
figure below. Find the dimensions that maximize the enclosed area. (Hint: First express y as a function of
x, and then express A as a function of x.)
3. ) A page of a book is to have an area of 90 in2, with 1-in. margins at the bottom and sides and a ½ in.
margin at the top. Find the dimensions of the page that will allow the largest printed area.
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4.) A builder intends to construct a storage shed having a volume of 900 ft3, a flat roof, and a rectangular
base who width is three-fourths the length. The cost per square foot of the materials is $4 for the floor,
$6 for the sides, and $3 for the roof. What dimensions will minimize the cost?
5.) A pipeline for transporting oil will connect two points A and B that are 3 miles apart and on opposite
banks of a straight river 1 mile wide. Part of the pipeline will run under water from A to C on the opposite
bank, and then above ground from C to B. If the cost per mile of running the pipeline under water is four
times the cost of running it above ground, find the location of C that will minimize the cost (Disregard the
slope of the riverbed).
6.) Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by
cutting equal squares from the corners and turning up the sides.
x
x
x
x
x
x
x
24 - 2x
x
2
7.) A small island is 2 miles from the nearest point P on the straight shoreline of a large lake. If a woman on
the island can row a boat 3 miles per hour and can walk 4 miles per hour, where should the boat be
landed in order to arrive at a town 10 miles down the shore from P in the least time?
2 miles
P x
10 - x
town
8.) A rectangle has two corners on the x-axis and the other two on the parabola y  12  x 2 , with y  0 .
What are the dimensions of the rectangle of this type with maximum area?
(x, 12 - x2)
9.) A wire of length 100 centimeters is cut into two pieces; one is bent to form a square, the other an
equilateral triangle. Where should the cut be made if (a) the sum of the two areas is to be a minimum;
(b) a maximum? (Allow the possibility of no cut.)
100 - 3x
100 - 3x
4
100 - 3x
3x
x
x
x
4
3
x
cents per mile if the truck travels at x miles per hour. In
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addition, the driver gets $12 per hour. What is the most economical speed at which to operate the truck
on a 400-mile run if the highway speed is required to be between 40 and 55 miles per hour inclusive?
10.) The operating cost of a certain truck is 25 
11.) Find the points on the parabola x  2y 2 that are closest to the point (10, 0). Hint: Minimize the square of
the distance (AB) between (x, y) and (10, 0).
(2y 2, y) A
B (10, 0)
12.) Suppose that Farmer Brown has 180 feet of fence and wants the pen to adjoin to the whole side of the
barn, as shown below. What should the dimensions be for maximum area? Note that 0  x  40 in this
case.
100 feet
BARN
PEN
x
y
4
Assignment 5.2
For 1 and 2, an object is moving along a horizontal number line so that its position s at any given time t is
given. The time interval is also given. Find the velocity and acceleration at time t. Also give a motion
diagram.
1.) s t   2t 3  15t 2  24t  6 when 0  t  5
2.) s  t  
t 2  3t  1
on  2,2
t2  1
3.) An object is fired directly upward. Its distance in feet, s, above the ground at t seconds is
s t   144t  16t 2 .
a.) Find the velocity and acceleration at t seconds.
b.) Find the maximum height reached.
c.) Find the duration of the flight.
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4.) A particle moves on the x-axis in such a way that its position at time t is given by x   2t  1  t  1  .
2
a.) At what time(s) t is the particle at rest? Justify your answer.
b.) During what interval of time is the particle moving to the left? Justify your answer.
c.) At what time during the interval found in (b) is the particle moving most rapidly (that is, the speed is a
maximum)? Justify your answer.
d.) For t  0 , give a diagram representing the motion of the particle.
e.) For what times is the object speeding up?
5.) A particle moves along the x-axis so that at time t its position is given by x  t   sin  t 2  for 1  t  1 .
a.) Find the velocity at time t.
b.) Find the acceleration at time t.
c.) For what values of t does the particle change direction?
d.) Find all values of t for which the particle is moving to the left.
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6.) The picture below shows the velocity, v  f t  , of a particle moving on a coordinate line.
a.) When does the particle move forward?
2
Move backward?
5
-2
Speed up?
Slow down?
b.) When is the particle’s acceleration positive?
Negative?
Zero?
c.) When does the particle move at its greatest speed?
d.) When does the particle stand still for more than an instant?
e.) Graph the speed vs. time.
f.) Graph the acceleration vs. time.
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10
Assignment 5.3
1.) If V  5p
3
2
and
dV
dp
.
 4 when V  40 find
dt
dt
2.) Gas is being pumped into a spherical balloon at a rate of 5 cubic feet/minute. Find the rate at which the
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radius is changing when the diameter is 18 inches. V   r 3
3
(Hint 18 inches = 1.5 feet)
3.) A ladder 17 feet long leans against a vertical building. If the bottom of the ladder slides away from the
building horizontally at a rate of 3 ft/sec, how fast is the ladder sliding down the building when the top of
the ladder is 8 feet from the ground?
4.) A person flying a kite holds the string 5 ft above ground level, and the string is payed out at a rate of 2
ft/sec as the kite moves horizontally at an altitude of 105 ft. Assuming there is no sag in the string; find
the rate at which the kite is moving when 125 ft. of string has been payed out.
8
5.) A trough is 15 ft long and 4 ft across the top as shown in the figure. Its ends are isosceles triangles with a
height of 3 ft. Water runs into the trough at a rate of 2.5 cubic ft/minute. How fast is the water level
rising when it is 2 ft deep?
15 ft
4 ft
3 ft
6.) The area of an equilateral triangle is decreasing at a rate of 4 square cm/min. Find the rate at which the
length of a side is changing when the area of the triangle is 100 3 square cm.
7.) If a spherical tank of radius a contains water that has a maximum depth h, then the volume V of water in
1
the tank is given by V   h2  3a  h  . Suppose a spherical tank of radius 16 ft is being filled at a rate of
3
100 gal/min. Approximate the rate at which the water level is rising when h = 4 ft. 1gal  0.1337 ft 3 
8.) A girl starts at a point A and runs east at a rate of 10 ft/sec. One minute later, another girl starts at A and
runs north at a rate of 8 ft/sec. At what rate is the distance between them changing 1 minute (60
seconds) after the second girl starts?
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9.) If x 2  3y 2  2y  10 and
dx
dy
.
 2 when x = 3 and y = -1, find
dt
dt
10.) As sand leaks out of a hole in a container, it forms a conical pile whose altitude is always the same as its
radius. If the height of the pile is increasing at a rate of 6 inches/minute, find the rate at which the sand is
leaking out when the altitude is 10 inches.
11.) A man on a dock is pulling in a boat using a rope attached to the bow of the boat 1 ft above water level
and passing through a simple pulley located on the dock 8 ft above water level. If he pulls in the rope at a
rate of 2 ft/sec how fast is the boat approaching the dock when the bow of the boat is 24 ft from a point
that is directly below the pulley?
12.) An airplane is flying at a constant speed of 360 miles/hr and climbing at an angle of 45 degrees. At the
moment the plane’s altitude is 10,560 ft, it passes directly over an air traffic control tower on the ground.
Find the rate at which the airplane’s distance from the tower is changing 1 minute later (neglect the
height of the tower) (10,560 ft = 2 miles)
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13.) A missile is fired vertically from a point that is 5 miles from a tracking station and at the same elevation.
For the first 20 sec of flight, its angle of elevation  changes at a constant rate of 2 degrees/sec. Find the



velocity of the missile when the angle of elevation is 30 degrees.  2  rad 
90


14.) The speed of sound in air at 0 degrees C or (273 degrees K) is 1087 ft/sec, but this speed increases as the
temperature rises. If T is temperature in degrees K, the speed of sound v at this terperature is given by
v  1087 T / 273 . If the temperature increases at a rate of 3 degrees K per hour, approximate the rate at
which the speed of sound is increasing when T = 30 degrees C (or 303 degrees K).
15.) Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10 cubic inches/min.
(a) How fast is the level in the pot rising when the coffee in the cone is 5 inches deep?
(b) How fast is the level in the cone falling at that moment?
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Assignment 5.4
1.) Find the approximate value of f  x   x at x  145 from the tangent to the graph at x  144 .
2.) Find the approximate value of f  x   4 x at x  17 from the tangent to the graph at x  16 .
3.) Find the approximate value of f  x   9  tan x at x  0.18 from the tangent to the graph at x  0 .
4.) If g  x  
x
, find the linear approximation of g 2.27 at x  2 .
x 1
5.) Let f be a function defined by f  x   1  tan x 
3
for 

x

.
4
2
(a) Write an equation for the line tangent to the graph of f at the point where x  0 .
2
(b) Using the equation found in part (a), approximate f  0.02 .
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6.) Consider the curve defined by 8 x 2  5xy  y 3  149
dy
(a) Find
.
dx
(b) Write an equation for the line tangent to the curve at the point (4, -1).
(c) There is a number k so that the point (4.2, k) is on the curve. Using the tangent line found in part (b),
approximate the value of k.
(d) Write an equation that can be solved to find the actual value of k so that the point (4.2, k) is on the
curve.
(e) Solve the equation found in part (d) for the value of k.
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x2
7.) Line l is tangent to the graph of y  x 
at the point Q, as shown in the figure below.
500
ℓ
yx
Q
x2
500
(0, 20)
P
O
(a) Find the x-coordinate of point Q.
(b) Write an equation for the line ℓ.
x2
shown in the figure, where x and y are measured in feet, represents
500
a hill. There is a 50 foot tree growing vertically at the top of the hill. Does a spotlight at point P
directed along line ℓ shine on any part of the tree? Show work.
(c) Suppose the graph of y  x 
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8.) Graph f ''  x   1  x sin x ln x for 3  x  9 , using your calculator. Label coordinates of endpoints, high
points, low points and zeros. Round all values to the nearest thousandth.
(a) For what values of x is f concave up?
(b) Find the x–values of all local minimum values of f ' .
(c) Find the x-coordinates of all points of inflection of f ' .
9.) Use your calculator to find all the solutions of the equation sin x  x cos x  cos x between -3 and 3.
Round to the nearest thousandth.
10.) Give the point of intersection of the graphs of y  sin2x and y  6 x  6 . Round the coordinates to the
nearest thousandth.
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Unit 5 Review
In 1-4, f  x   x 4  3 x  2 and g  x   4sin x  5 . Round each answer to the nearest thousandth.
1.) Find all zeros of f.
2.) Find the coordinates of the points of intersection of f and g.
3.) Find the coordinates of the lowest point on f.
4.) Find the slope of the tangent to
f
at x  2 .
g
5.) A child is flying a kite. If the kite is 90 feet above the child’s hand level and the wind is blowing it at a
horizontal course at 5 ft/sec, how fast is the child paying out cord when 150 feet of cord is out? (Assume
the cord forms a line – actually an unrealistic assumption)
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6.) Two ships sail from the same island port, one going north at 24 knots (24 nautical miles per hour) and the
other east at 30 knots. The northbound ship departed at 9:00 A.M. and the eastbound ship departed at
11:00 A.M. How fast is the distance between them increasing at 2:00 P.M.?
7.) A student is using a straw to drink from a conical paper cup, whose axis is vertical, at a rate of 3 cm³/sec.
If the height of the cup is 10 cm and the diameter of its opening is 6 cm, how fast is the level of the liquid
falling when the depth of the liquid is 5 cm?
8.) A light in a lighthouse 1 kilometer offshore from a straight shoreline is rotating at 2 revolutions/minute.
How fast is the beam of light moving along the shoreline when it passes the point ½ kilometer from the
point opposite the lighthouse?
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9.) Andy, who is 6 feet tall, is walking away from a street light pole 30 feet high at a rate of 2 ft/sec.
(a) How fast is his shadow increasing in length when Andy is 24 ft from the pole? 30 ft?
(b) How fast is the tip of his shadow moving?
(c) To follow the tip of his shadow, at what angular rate must he lift his head when his shadow is 6 ft
long?
4
10.) The radius r of a sphere is increasing at a constant rate of 0.04 cm/sec. V   r 3
3
(a) At the time when the radius of the sphere is 10 cm, what is the rate of increase of its volume?
(b) At the time when the volume of the sphere is 36 cubic cm, what is the rate of increase of the area of
a cross section through the center of the sphere?
(c) At the time when the volume and the radius of the sphere are increasing at the same numerical rate
what is the radius?
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11.) The balloon shown in the picture below is in the shape of a cylinder with hemispherical ends of the same
radius as that of the cylinder. The balloon is being inflated at the rate of 261 cubic cm/min. At the
instant the radius of the cylinder is 3 cm, the, the volume of the balloon is 144 cubic cm and the radius
4
of the cylinder is increasing at the rate of 2 cm/min. Vcylinder   r 2h and Vsphere   r 3
3
(a) At this instant what is the height of the cylinder?
(b) At this instant, how fast is the height of the cylinder increasing?
12.) A ladder 15 ft long is leaning against a building and the base of the ladder is moving away from the
building at a constant rate of ½ ft/sec.
(a) Find the rate in ft/sec at which the top of the ladder is sliding down the building when the base is 9 ft
from the building.
(b) Find the rate of change in square ft/sec the area of the triangle created by the ladder, the building and
the ground creates when the base of the ladder is 9 feet from the building.
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Multiple Choice
13.) The approximate value of y  4  sin x at x  0.12 obtained from the tangent line to the graph at x  0
is?
(a) 2.00
(b) 2.03
(c) 2.06
(d) 2.12
(e) 2.24
14.) Let f be a differentiable function such that f  3  2 and f '  3  5 . If the tangent line to the graph of f at
x  3 is used to find an approximation to a zero of f, that approximation is?
(a) 0.4
(b) 0.5
(c) 2.6
(d) 2.12
(e) 2.24
15.) For the function f, f '  x   2x  1 and f 1  4 . What is the approximation for f 1.2  found by using the
tangent line to the graph of f at x  1 ?
(a) 0.6
(b) 3.4
(c) 4.2
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(d) 4.6
(e) 4.64
16.) The sides and diagonal of the rectangle below are strictly increasing with time. At the instant when x  4
dx dz
dy
dz
and y  3 ,
and

 k . What is the value of k at that instant?
dt dt
dt
dt
(a)
1
4
(b)
1
3
(c) 3
(d) 4
(e) Cannot be determined
17.) The radius of a circle is increasing. At a certain instant, the rate of increase in the area of the circle is
numerically equal to twice the rate of increase in its circumference. What is the radius of the circle at
that instant?
(a)
1
2
(b) 1
(c)
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2
(d) 2
(e) 4