Optimization Problems
Calculus BC
Note: To complete the optimization problems, you should have learned how to use a system of
equations, the substitution method, and your calculator to find the maximum or minimum value.
These are sometimes called quadratic word problems.
Ex 1: Find two positive numbers whose product is 115 and whose sum is a minimum.
Eqn 1: xy  115
Eqn 2: S  x  y
where x represents my first number, y represents my second number, and S represents the
sum of the two numbers.
115
Combine equations by solving for x or y in eqn 1, you get: S  x 
x
Put S into Y1 and use calc to find the minimum x-value.
The minimum value is (10.724, 21.448)
Use the x-value of the minimum and substitute back into Eqn 1: (10.724)y=115
Solve for y.
Solution: x = 10.724
y = 10.724
Write a function for each problem, and use your graphing calculator to solve. Give decimal
answers correct to three decimal places. Be sure to sketch the graph you used, and label it with
your answer.
1. Find two positive numbers such that their product is 192 and their sum is a minimum.
2. Find two positive numbers such that their product is 192 and the sum of the first plus
three times the second is a minimum.
3. A rancher has 200 feet of fencing with which to enclose two adjacent
rectangular corrals, as shown. What dimensions should be used so
that the enclosed area will be a maximum?
4. A manufacturer wants to design an open box having a square base and a surface area of 108
square inches. What dimensions will produce a box with maximum volume?
5. Which points on the graph of
are closest to the point (0, 2)?
6. A rectangular page is to contain 24 square inches of print. The margins at the top and the
bottom of the page are to be 1.5 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?
7. Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be
stayed (supported, braced, fixed) 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?
8 Four feet of wire is to be used to form a square and a circle. How much of the wire should be
used for the square and how much should be used for the circle to enclose the maximum total
area?
9. A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a
straight river. He needs no fence along the river. What are the dimensions of the field that has
the largest area?
10. Find the area of the largest rectangle that can be inscribed in a semicircle of radius r. (Hint: r
is a constant. Also, for a given x coordinate, the corresponding y coordinate of a point on the
semicircle is given by
√