Calculus and Inequalities in Word Problems

Calculus applications
Exercise 6.1

23. Manufacturing A company manufacturing two types of leaf blowers, an electric Turbo model and gas-powered Tornado model. The company's production plan calls for the production of at least 780 blowers per month.
a. Write the inequality that describes the production plan, if x represents the number of Turbo blowers and y represents the number of Tornado blowers.
b. Graph the region determined by this inequality in the context of the application.

25. Sales Trix Auto sales sells used cars. To advertise its cars to target groups, they advertise with x 1-minute spots on cable television, at a cost of $240 per minute, and y 1-minute spots on radio, at a cost of $150 per minute. Suppose the company has at most $36,000 to spend on advertising.
a. Write the inequality that describes this constraint on advertising.
b. Graph the region determined by this inequality I the context of the application.

27. Politics A political candidate wishes to use a combination of x newspaper and y television advertisements in his campaign. Each 1-minute ad on television reaches 0.12 million eligible voters, and each 1-minute ad on radio reaches 0.009 million legible voters. The candidate feels that he must reach at least 7.56 million eligible voters and that he must buy at least 100 minutes of investments.
a. Write the inequalities that describes these advertising requirements.
b. Graph the region determined by these inequalities in the context of the application.

31. Production A firm produces three different-size television sets on two assembly lines. The following table summarizes the production capacity of each assembly line and the minimum number of each size TV needed to fill orders.
a. Write the inequalities that are the constraints.
b. Graph the region that satisfies the inequalities and identify the corners.

Assembly Line 1 Assembly Line 2 Number Ordered
19-in.TV 80 per day 40 per day 3200
25-in.TV 20 per day 20 per day 1000
35-in.TV 100 per day 40 per day 3400

35. Manufacturing Easy boy manufactures two types of chairs, Standard and Deluxe. Each Standard chair requires 4 hours to construct and finish, and each Deluxe chair requires 6 hours to construct and finish. Upholstering takes 2 hours for a standard chair and 6 hours for a Deluxe chair. There are 480 hours available each day for construction and finishing and there are 300 hours available per day for upholstering.
a. Write the inequalities that describe the application.
b. Graph the solution of the system of inequalities and identify the corners of the region.

Exercise 6.2

19. Manufacturing A company manufactures two types of leaf blowers, an electric Turbo model and gas powdered Tornado model. The company's production plan calls foe the production of at least 780 blowers per month. It costs $78 to produce each Turdo model and $117 to manufacture each Tornado model, and the company has at most $76,050 per month to use for the production. Find the number of units that should be produced to maximize the profit for the company if the profit on each Trobo model is $32 and the profit on each Tornado model is $45.

21. Manufacturing Safeco Company produces two types of chainsaws, the Safe cut and the Safe cut Deluxe. The Safe cut model requires 2 hours to assemble and 1 hour to paint, and the Dulexe model requires 3 hours to assemble and ½ hour to paint. The daily maximum numbers of hours available for assembly is 36, and the daily maximum number of hours available for painting is 12. If the profit is $24 per units on the safe cut model and $30 per units per the Dulexe model, how many units of each type will maximum the daily profit?

25. Production A firm produces three different-size television sets on two assembly lines. The following table summaries the production capacity of each assembly line, the number of each size TV ordered by a retailer, and the daily operating costs for each assembly line. How many days should each assembly line run to fill this order with minimum cost? What is the minimum cost?

Assembly Line 1 Assembly Line 1 Number Ordered
19-in.Tv 80 per day 40 per day 3200
27-in.TV 20 per day 20 per day 1000
35-in.TV 100 per day 40 per day 3400
Daily Cost($) 20,000 40,000

31. Nutrition In a hospital war, patients are grouped into two general nutritional categories depending on the amount of solid food in their diet, and are provided two different diets with different amounts of the solid foods and with some detrimental substances. Table below gives the patients groups, the weekly diet requirements for each group, ant the amount of detrimental substance in each diet. How many serving of each diet will satisfy the nutritional requirements and minimize the detrimental substance? What is the minimum amount of detrimental substance?

Diet A (oz/serving) Diet B (oz/serving) Minimum Daily requirement (oz)
Group 1 2 1 18
Group 2 4 1 26
Detrimental .09 .035