
College Calculus (Conway)
Name___________________________________
Optimization Problems
Date________________
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Solve each optimization problem.
1) A supermarket employee wants to construct an open-top box from a  by  in piece of
cardboard. To do this, the employee plans to cut out squares of equal size from the four
corners so the four sides can be bent upwards. What size should the squares be in order to
create a box with the largest possible volume?
Illustration of problem
 in
 in
V = the volume of the box x = the length of the sides of the squares
Function to maximize: V( x)( x) x where  x

Sides of the squares:
in

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Worksheet by Kuta Software LLC
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2) A cryptography expert is deciphering a computer code. To do this, the expert needs to
minimize the product of a positive rational number and a negative rational number, given that the
positive number is exactly  greater than the negative number. What final product is the expert
looking for?
Illustration of problem
No illustration necessary for this problem
P = the product of the two numbers x = the positive number
Function to minimize: P x( x) where   x 

Smallest product of the two numbers: 

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Worksheet by Kuta Software LLC
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3) A farmer wants to construct a rectangular pigpen using  ft of fencing. The pen will be built
next to an existing stone wall, so only three sides of fencing need to be constructed to enclose
the pen. What dimensions should the farmer use to construct the pen with the largest possible
area?
Illustration of problem
A = the area of the pigpen x = the length of the sides perpendicular to the stone wall
Function to maximize: A x( x) where  x
Dimensions of the pigpen:  ft (perpendicular to wall) by  ft (parallel to wall)
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Worksheet by Kuta Software LLC
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4) A rancher wants to construct two identical rectangular corrals using  ft of fencing. The
rancher decides to build them adjacent to each other, so they share fencing on one side. What
dimensions should the rancher use to construct each corral so that together, they will enclose the
largest possible area?
Illustration of problem
A = the total area of the two corrals x = the length of the non-adjacent sides of each corral
 x
Function to maximize: A x
where  x



Dimensions of each corall:
ft (non-adjecent sides) by
ft (adjacent sides)


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Worksheet by Kuta Software LLC
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5) A company has started selling a new type of smartphone at the price of $ x where x
is the number of smartphones manufactured per day. The parts for each smartphone cost $
and the labor and overhead for running the plant cost $ per day. How many smartphones
should the company manufacture and sell per day to maximize profit?
Illustration of problem
No illustration necessary for this problem
p = the profit per day x = the number of items manufactured per day
Function to maximize: p x( x)( x) where  x 
Optimal number of smartphones to manufacture per day: 
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Worksheet by Kuta Software LLC
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6) Engineers are designing a box-shaped aquarium with a square bottom and an open top. The
aquarium must hold  ft³ of water. What dimensions should they use to create an acceptable
aquarium with the least amount of glass?
Illustration of problem
x
x
A = the area of the glass
x = the length of the sides of the square bottom

Function to minimize: A x   x  where  x 
x
Dimensions of the aquarium:  ft by  ft by  ft tall
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Worksheet by Kuta Software LLC
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7) A geometry student wants to draw a rectangle inscribed in the ellipse x   y  . What is
the area of the largest rectangle that the student can draw?
Illustration of problem
y





















 x
Example rectangle not optimal






A = the area of the rectangle
x = half the base of the rectangle
Function to maximize: A x
 x 
where  x

Area of largest rectangle: 
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Worksheet by Kuta Software LLC
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8) Which points on the graph of y x  are closest to the point (, )?
Illustration of problem

y




















 x
d = the distance from point (, ) to a point on the parabola
x  ( x  )
x = the x-coordinate of a point on the parabola

where   x 
 
Points on the parabola that are closest to the point (, ): 
, ,
 
Function to minimize: d
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(
-8-
)(
 
,
 
)
Worksheet by Kuta Software LLC
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9) A geometry student wants to draw a rectangle inscribed in a semicircle of radius . If one side
must be on the semicircle's diameter, what is the area of the largest rectangle that the student
can draw?
Illustration of problem

y









Example rectangle not optimal






A = the area of the rectangle



 x
x = half the base of the rectangle
Function to maximize: A x    x  where  x
Area of largest rectangle: 
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Worksheet by Kuta Software LLC
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10) A graphic designer is asked to create a movie poster with a  in² photo surrounded by a  in
border at the top and bottom and a  in border on each side. What overall dimensions for the
poster should the designer choose to use the least amount of paper?
Illustration of problem
 in
 in
 in
 in
A = the area of the poster
x = the width of the photo

Function to minimize: A( x)
  where  x 
x
Dimensions of the entire poster:  in wide by  in tall
(
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)
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Worksheet by Kuta Software LLC
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11) Which point on the graph of y
x is closest to the point (, )?
Illustration of problem

y






















 x
d = the distance from point (, ) to a point on the curve
( x)  (
x = the x-coordinate of a point on the curve

x ) where   x 


Point on the curve that is closest to the point (, ):
,


Function to minimize: d
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(
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)
Worksheet by Kuta Software LLC
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12) Two vertical poles, one  ft high and the other  ft high, stand  feet apart on a flat field. A
worker wants to support both poles by running rope from the ground to the top of each post. If
the worker wants to stake both ropes in the ground at the same point, where should the stake
be placed to use the least amount of rope?
Illustration of problem
 ft
 ft
 ft
L = the total length of rope
x = the horizontal distance from the short pole to the stake
Function to minimize: L x     ( x)    where  x
Stake should be placed:  ft from the short pole (or  ft from the long pole)
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Worksheet by Kuta Software LLC
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13) An architect is designing a composite window by attaching a semicircular window on top of a
rectangular window, so the diameter of the top window is equal to and aligned with the width of
the bottom window. If the architect wants the perimeter of the composite window to be  ft,
what dimensions should the bottom window be in order to create the composite window with
the largest area?
Illustration of problem
A = the area of the composite window x = the width of the bottom window = the diameter of the top window


x x

x

Function to maximize: A x
  
 
where  x








Dimensions of the bottom window:
ft (width) by
ft (height)


(
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)
()
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