Applications Involving Inequalities Our last “new” stuff of the chapter!!! 2.9c.

Applications Applications Involving Involving

InequalitiesInequalitiesOur last “new” stuff of Our last “new” stuff of

the chapter!!! 2.9cthe chapter!!! 2.9c

Designing a Box – Revisited!!!Designing a Box – Revisited!!!Dixie Packaging Company has contracted with another firm todesign boxes with a volume of at least 600 cu. in. Squares arestill to be cut from the corners of a 20-in. by 25-in. piece ofcardboard, with the flaps folded up to make an open box. Whatsize squares should be cut from the cardboard?

Do you recall the diagram???

The equation for the volume???

25 2 20 2V x x x We need a volume of at least 600:

25 2 20 2 600x x x Solve graphically!!!

Squares with side lengths between 1.659 in. and 6.159 in.Squares with side lengths between 1.659 in. and 6.159 in.(inclusive) will produce a volume of 600 cu. in. or greater(inclusive) will produce a volume of 600 cu. in. or greater..

Designing a Juice Can – Revisited!!!Designing a Juice Can – Revisited!!!Stewart Cannery will package tomato juice in 2-liter cylindricalcans. Find the radius and height of the cans if the cans have asurface area that is less than 1000 square centimeters.

Do you recall the equations for volume and surface area???2 2000V r h

2 40002S r

r

The inequality to be solved:

2

2000h

r

22 2S r rh

2 40002 1000r

r Solve graphically!!!

4.619 9.655r


To find the height, let’s solve a double inequality:

4.619 9.655r 2

2000h

r

2 2 24.619 9.655r 2 2 24.619 9.655r

2 2 2

1 1 1

4.619 9.655r

2 2 2

2000 2000 2000

4.619 9.655r


To find the height, let’s solve a double inequality:

2

2000h

r

2 2 2

2000 2000 2000

4.619 9.655r

2 2

2000 2000

4.619 9.655h

29.839 6.829h The surface area of the can will be less than 1000 sq. cm.The surface area of the can will be less than 1000 sq. cm.

if its radius is between 4.619 cm and 9.655 cm and itsif its radius is between 4.619 cm and 9.655 cm and itsheight is between 6.829 cm and 29.839 cm.height is between 6.829 cm and 29.839 cm.

More Practice ProblemsMore Practice ProblemsPederson Electric Co. charges $25 per service call plus$18 per hour for home repair work. How long did anelectrician work if the charge was less than $100? Assumethe electrician rounds off time to the nearest quarter hour.

The electrician worked no more than 4 hours.The electrician worked no more than 4 hours.

Let x be the number of hours worked

Repair charge:18 25x , which must be less than $100:

18 25 100x 18 75x

4.167x

More Practice ProblemsMore Practice ProblemsA candy company finds that the cost of making a certaincandy bar is $0.13 per bar. Fixed costs amount to $2000per week. If each bar sells for $0.35, find the minimumnumber of candy bars that will earn the company a profit.

The company needs to sell 9091 candy bars to make a profitThe company needs to sell 9091 candy bars to make a profit

Let x be the number of candy bars made

Costs: 0.13 2000C x

0.35 0.13 2000x x

Income: 0.35I xNeed income to be greater than costs:

0.22 2000x 9090.909x

More Practice ProblemsMore Practice ProblemsThe total electrical resistance R of two resistors connectedin parallel with resistances R and R is given by1

One resistor has a resistance of 2.3 ohms. Let x be the resistance of the second resistor.1 2

1 1 1

R R R

2

(a) Express the total resistance R as a function of x.

1 1 1

2.3R x 2.3 2.3x Rx R

2.3 2.3x R x 2.3

2.3

xR

x



1 1 1

R R R

2

(b) Find the resistance in the second resistor if the totalresistance of the pair is at least 1.7 ohms.

2.31.7

2.3

x

x

2.3

1.7 02.3

x

x

2.3 1.7 2.3

02.3

x x

x

0.6 3.91

02.3

x

x



1 1 1

R R R

2

(b) Find the resistance in the second resistor if the totalresistance of the pair is at least 1.7 ohms.

0.6 3.91

2.3

xf x

x

Where is this function greater thanor equal to zero? Check the graph!

The resistance of the second resistor is at least 6.517 ohmsThe resistance of the second resistor is at least 6.517 ohms

Applications Involving Inequalities Our last “new” stuff of the chapter!!! 2.9c.

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