Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet (Page 28 of 41) in pdf

Worksheet # 21: Optimization

1. Suppose that f is a function on an open interval I = (a, b) and c is in I. Suppose that f is continuous

at c, f (x) > 0 for x > c and f (x) < 0 for x < c. Is f (c) an absolute minimum value for f on I?

Justify your answer.

2. Find the dimensions of x and y of the rectangle of maximum area that can be formed using 3 meters

of wire.

(a) What is the constraint equation relating x and y?

(b) Find a formula for the area in terms of x alone.

3. A farmer has 2400 feet 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?

4. A hockey team plays in an arena with a seating capacity of 15000 spectators. With the ticket price set

at $12, average attendance at a game has been 11000. A market survey indicates that for each dollar

the ticket price is lowered, average attendance will increase by 1000. How should the owners of the

team set the ticket price to maximize their revenue from ticket sales?

5. An oil company needs to run a pipeline to a nearby station. The station and oil company are on

opposite sides of a river that is 1 km wide, and that runs exactly west-east and the station is 10 km

east along the river from the the oil company. The cost of building pipe on land is $200 per meter

and the cost of building pipe in water is $300 per meter. Set up an equation whose solution(s) are the

critical points of the cost function for this problem.

Find the least expensive way to construct the pipe.

6. A ﬂexible tube of length 4 m is bent into an L-shape. Where should the bend be made to minimize

the distance between the two ends?

7. A 10 meter length of rope is to be cut into two pieces to form a square and a circle. How should the

rope be cut to maximize the enclosed area?

8. Find the point(s) on the hyperbola y =

that is (are) closest to (0, 0). Be sure to clearly state what

function you choose to minimize or maximize and why.

9. Consider a can in the shape of a right circular cylinder. The top and bottom of the can is made of

a material that costs 4 cents per square centimeter, and the side is made of a material that costs 3

cents per square centimeter. We want to ﬁnd the dimensions of the can which has volume 72 π cubic

centimeters, and whose cost is as small as possible.

(a) Find a function f (r) which gives the cost of the can in terms of radius r. Be sure to specify the

domain.

(b) Give the radius and height of the can with least cost.

10. A box is to have a square base, no top, and a volume of 10 cubic centimeters. What are the dimensions

of the box with the smallest possible total surface area? Provide an exact answer; do not convert your

answer to decimal form. Make a sketch and introduce all the notation you are using.

Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 28