Section 5.6 Optimization and Modeling

Strategy for Solving Optimization Problems:

1. First, read the problem carefully, looking for important information (Do not read the problem like

a novel).

2. Draw a picture! Label your figure with appropriate variables.

3. Write down formulas for the quantity to be maximized (or minimized) and for the quantity to be

constrained in terms of your assigned variables.

4. Use your constraint formula, by solving for one variable in terms of the other, to determine the

function f(x) to be maximized (or minimized) on the interval I.

5. Using previously learned methods, find the absolute maximum (or minimum) value of f(x) on

the interval I and the value(s) of x where this occurs.

1. Find two non-negative numbers whose sum is 44 and whose product is a minimum.

2. Find two positive numbers x and y with xy = 300 such that the sum x+ 3y is a minimum.

3. ABC Daycare wants to build a fence to enclose a rectangular playground. The area of the

playground is 910 square feet. The fence along three of the sides costs $5 per foot and the fence

along the fourth side, which will be made of brick, costs $15 per foot. Find the length of the

brick fence that will minimize the cost of enclosing the playground. (Round your answer to one

decimal place.)

4. A rancher wants to create two rectangular pens, as shown in the figure, using an existing fence line

as one side. If there are 510 feet of fence available, what dimensions should be used to maximize

the total area of the pens?

2 Fall 2016, Maya Johnson

5. You are building a right-angled triangular flower garden along a stream as shown in the figure.

The fencing of the left border costs $10 per foot, while the fencing of the lower border costs $2

per foot. (No fencing is required along the river.) You want to spend $600 and enclose as much

area as possible. What are the dimensions of your garden, and what area does it enclose? [The

area of a right-triangle is given by A = xy/2.]

6. A box with a square base and open top must have a volume of 32, 000cm3. Find the dimensions

of the box that minimize the amount of material used.

3 Fall 2016, Maya Johnson

7. Bob wants to create two pens, as shown in the figure. One pen is for a garden and it needs a

heavy duty fence to keep out the critters. This heavy duty fence costs $6 per foot. The dog pen

shares a side with the garden and has a lighter weight fence on the other three sides that costs $3

per foot. If each pen is to have an area of 1, 920, find the values of x and y that would minimize

the total cost of the fencing.

8. If 30, 000cm2 of material is available to make a box with a square base and an open top, find the

largest possible volume of the box.

4 Fall 2016, Maya Johnson

SAax2t4xy.3oooo-4xy-3oo@o-x2-yi5f0-xgVzX2y-X2fsjot.xqf.7V-75oox-xy1Vk25oo-3yx2-ottnsi75oofsyIVjtIj.fgt.o56900€30000=3×-2 Amax

=ooo=×2 May Volume× - " 0

V( 160 )= -75001100) -11004=500000

9. A baseball team plays in a stadium that holds 58, 000 spectators. With ticket prices at $10,

the average attendance had been 49, 000. When ticket prices were lowered to $8, the average

attendance rose to 51, 000.

(a) Find the demand function (price p as a function of attendance x), assuming it to be linear.

(b) How should ticket prices be set to maximize revenue? (Round your answer to the nearest

cent.)

10. A company is going to make open-topped boxes out of 17 ⇥ 18-inch rectangles of cardboard by

cutting squares out of the corners, shown blue in the left figure, and folding up the sides. The

finished box is the right picture. What is the largest volume box the company can make this way?

(Round your answer to one decimal place.)

5 Fall 2016, Maya Johnson

UseSTATtEDIT-STAT-CAK-LinReg4LzFfoftE.pz-ao1xt-9oExt5800cRlXk-oolx2t59xR1lx1-s.oozxt59t0.OO2X-5I.ooz-7X-295ooRH-aozTsYh@RY295oGLOnmaX.Max

price pa - .oOl(295007+59=29.5

.

x x

x x

:£*2× ×

' ' '2××*2× µhf ;× x

¥t÷5*ke?exE*h÷¥ rite.EE:396.3M€

a Max VolumeV '=( 18-4×1/12 -2×7248×-2×2)

⇐306-104×+8×2 - 36×+4×2 V( 2.91=(2.9118-212.9)/(17-212.9) )✓ '= 12×2-140×+306=0

= 396.3

X=14O±~402-4112)( 306 )÷2)

Xa 140+70=4*140-70.124

11=8.8 or 2.9

8.8 is too big

So 11=2.9