Optimization Practice
Name: __________________________________
Score: ________
Math 1210 – 001 Summer 2012
Instructor: Katrina Johnson
Clearly indicate the meaning of each variable as you set up each problem,
draw a diagram if necessary. Answer each part of the following questions to
complete each problem. Feel free to use a calculator. Show all of your work.
Don’t forget units. This assignment is due: Monday, July 23. I will not
accept late assignments!
(4 pts) 1. During the summer months Terry makes and sells necklaces on the beach. Last
summer he sold the necklaces for $10 each and his sales averaged 20 necklaces per day.
When he increased the price by $1, he found that he lost two sales per day. So the price of
a necklaces is represented by the function,
by the function,
1
k(x) = − x + 20 , and the revenue is modeled
2
1
R(x) = − x 2 + 20x , where x represents the number of necklaces sold.
2
The material for each necklace cost Terry $6. This is represented by the cost function,
€ number of necklaces produced. What should the selling
C(x) = 6x , where x represents then
price be to maximize Terry’s profit? Hint: Recall the profit function is P(x) = R(x) − C(x) .
€
i)
ii)
€
You would like to minimize/maximize (circle one) _____________.
What is the objective function?
€
iii)
What is the derivative of the objective function?
iv)
What are the critical values of the objective function? At which
critical values do you minimize/maximize the objective function?
How do you know?
v)
What should the selling price be to maximize the profit?
(4 pts) 2. A dairy farmer plans to enclose a rectangular pasture adjacent to
a river. To provide enough grass for the herd, the pasture must contain
180,000 square meters. No fencing is required along the river. What
dimensions use the least amount of fencing?
i)
ii)
You would like to minimize/maximize (circle one) _____________.
a) What is the objective function? ______________________
b) What are the constraints? _________________________
c) Solve the constraints for one variable and substitute to get the
objective function as a function in one variable.
iii)
What is the derivative of the objective function?
iv)
What are the critical values of the objective function? At which
critical values do you minimize/maximize the objective function?
How do you know?
v)
What dimensions for the pasture use the least amount of fencing?
(4 pts) 3. A manufacturer wants to design an open box that has a square
base and a surface area of 108 square inches. What dimensions will produce
a box with a maximum volume?
i)
ii)
You would like to minimize/maximize (circle one) _____________.
a) What is the objective function? ______________________
b) What are the constraints? _________________________
c) Solve the constraints for one variable and substitute to get the
objective function as a function in one variable.
iii)
What is the derivative of the objective function?
iv)
What are the critical values of the objective function? At which
critical values do you minimize/maximize the objective function?
How do you know?
v)
What dimensions will produce a box with a maximum volume?
(4 pts) 4. A soft drink can is to contain 12 fluid ounces. Assume the can is
made by forming sheet aluminum into a cylindrical container (open on one
end) and then attaching a circular lid. The bottom and sides cost $0.02 per
square meter, and the top costs $0.05 per square meter. What is the ratio
of height to base diameter of the can that is cheapest to manufacture?
( 1 fluid ounce = 2.957x10 −5 m 3 )
€
i)
ii)
You would like to minimize/maximize (circle one) _____________.
a) What is the objective function? ______________________
b) What are the constraints? _________________________
c) Solve the constraints for one variable and substitute to get the
objective function as a function in one variable.
iii)
What is the derivative of the objective function?
iv)
What are the critical values of the objective function? At which
critical values do you minimize/maximize the objective function?
How do you know?
v)
What is the ratio of height to base diameter of the can that is
cheapest to manufacture?
(4 pts) 5. Find the dimensions of the rectangle of largest area that has its
base on the x-axis and its other two vertices above the x-axis and lying on
the parabola y=8-x^2.