Warm-Up
1) You have 100 feet of fence and want to make a rectangular
pen Ms. Cojohn’s dog Tobin. What are the dimensions of the
pen that give Tobin the most room to play?
2) Find two positive numbers whose product is 12 and whose sum
is a minimum.
Students will be able to solve applied minimum
and maximum problems.
One of the most common applications of calculus
involves determining the maximum or minimum values
of a function. While we may be able to easily
find a max or min using calculus, setting up the
problem might be more difficult. Here are some
steps to help –
1) Carefully read the problem and identify
given quantities and quantities to be
determined.
2) Make a sketch & label it
3) Write down all equations related to your
problem/diagram.
a. One equation is a primary equation
b. One equation is the constraint equation
4) Use the constraint equation to solve for one
of the variables.
5) Substitute this equation into the primary equation.
6) Make sure your primary equation only has ONE variable, then
differentiate
7) Set the derivative equal to zero & solve!
8) Check your work with the 1st or 2nd Derivative Test
Example 1: A manufacturer wants to design an open box having a
square base and a surface area of 108 in2. What dimensions will
produce a box with maximum volume?
Example 2: A steel drum in the shape of a right circular cylinder
is required to have a volume of 100 ft3. A manufacturer is
required to make the drum with the least amount of metal. What
dimensions produce a drum with the minimum material?
Example 3: A rectangle has one corner on the graph of y  16  x 2 ,
another at the origin, a third on the positive y-axis, and the
fourth on the positive x-axis. For what value of x is the area
the largest?
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2
Example 4: Which points on the graph of y  4  x are the closest
to the point (0,2)?
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Name:
OPTIMIZATION PROBLEMS
1) Joe needs to enclose a rectangular field next to a building with a fence. He has
500 feet of fencing material. One side of the enclosed area is a building and won’t
need any fencing. Determine the dimensions of the field that will enclose the
largest area.
2) You want to construct a box whose base length is 3 times the base width. The
material used to build the top and bottom cost $10/ft2 and the material used to
build the sides cost $6/ft2. If the box must have a volume of 50ft3. Determine the
dimensions that will minimize the cost to build the box.
3) A manufacturer needs to make a cylindrical can that will hold 1.5 liters (equal to
1500 cubic centimeters) of liquid. Determine the dimensions of the can that will
minimize the amount of material used in its construction.
4) We want to construct a box with a square base and we only have 10 m2 of
material to use in construction of the box. Assuming that all the material is used
in the construction process determine the maximum volume that the box can
have.
5) A printer need to make a poster that will have a total area of 200 in2 and will
have 1 inch margins on the sides, a 2 inch margin on the top and a 1.5 inch
margin on the bottom. What dimensions will give the largest printed area?
6) We have a piece of cardboard that is 14 inches by 10 inches and we’re going to
cut out the corners as shown below and fold up the sides to form a box.
Determine the height of the box that will give a maximum volume.
Warm-Up:
1) A supermarket employee wants to construct an open-top box from a 14 by 30 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?
2) Which point on the graph of y  x is closest to the point (7, 0)?
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Objective: Students will be able to solve applied minimum and maximum problems.
To Solve -
Example 1: Jose has 5000 feet of fence. He would like to make a rectangular pen for his dog to play in his backyard. He would
like his pen to have the most space for his dog to play. What is the maximum area that can be obtained with 5000 feet of fence?
Example 2: A rectangular page is to contain 24 in2 of print. The margins at the top and bottom of the page are to be 1½ inches
and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that the least amount of
paper is used?
Example 3: Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires,
attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least
amount of wire?
Example 4: A cylinder is inscribed in a sphere with radius 3 inches. What is the volume of the cylinder with the greatest possible
volume?
Example 5: A piece of wire 40 cm long is to be cut into two pieces. One piece will be bent to form a circle; the other will be bent
to form a square. Find the lengths of the two pieces to cause the sum of the area of the circle and the area of the square to be a
minimum. How could you make the total area of the circle and the square a maximum?