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Question: A car maintenance shop must decide how many oil changes and how many tune-ups can be scheduled in a typical week. The oil change takes 20 min, and the tune-up requires 100 min. The maintenance shop makes a profit of $15 on an oil change and $65 on a tune-up. What mix of services should the shop schedule if the typical week has 8000 min available for these two types of services? How, if at all, do the maximum profit and optimal production policy change if the shop is required to schedule at least 50 oil changes and 20 tune-ups?Answer: Schedule 400 oil changes and no tune-ups; schedule 300 oil changes and 20 tune-ups.How do you set up the chart and figure everything out using the following steps? Make a mixture chart for the problem.Using the mixture chart, write the profit formula and the resource- and minimum-constraint inequalities.Draw the feasible region for those constraints and find the coordinates of the corner points.Evaluate the profit information at the corner points to determine the production policy that best answers the question.

Question

Question: A car maintenance shop must decide how many oil changes and how many tune-ups can be scheduled in a typical week. The oil change takes 20 min, and the tune-up requires 100 min. The maintenance shop makes a profit of $15 on an oil change and $65 on a tune-up. What mix of services should the shop schedule if the typical week has 8000 min available for these two types of services? How, if at all, do the maximum profit and optimal production policy change if the shop is required to schedule at least 50 oil changes and 20 tune-ups?

Answer: Schedule 400 oil changes and no tune-ups; schedule 300 oil changes and 20 tune-ups.

How do you set up the chart and figure everything out using the following steps? 

  1. Make a mixture chart for the problem.
  2. Using the mixture chart, write the profit formula and the resource- and minimum-constraint inequalities.
  3. Draw the feasible region for those constraints and find the coordinates of the corner points.
  4. Evaluate the profit information at the corner points to determine the production policy that best answers the question.
check_circleAnswer
Step 1

Here the oil changes take 20 minutes to make a profit of $15 and tune-up takes 100 minutes to make a profit of $65.

  1. Mixture is as follows.
Products
Profit
Oil change
20 minutes
$15
Tune-up change |100 minutes
$65
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Products Profit Oil change 20 minutes $15 Tune-up change |100 minutes $65

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Step 2
  1. Oil changes take 20 minutes and tune-up takes 100 minutes.

Altogether it takes 8000 for the week.

The constraints are,  

x20
20x 100 y 8000
Oil changes take 20 minutes to make a profit of $15
Then profit for oil is 15x.
Tune-up takes 100 minutes to make a profit of $65.
Then profit for oil is 65y
P 15x 65 y
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x20 20x 100 y 8000 Oil changes take 20 minutes to make a profit of $15 Then profit for oil is 15x. Tune-up takes 100 minutes to make a profit of $65. Then profit for oil is 65y P 15x 65 y

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Step 3

3.The feasible region of the constraints obtained in step 2 are d...

(0, 80)
80
60
20+100y 8000
40
20
(0,0)
300 (400, 0)
200
100
400
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(0, 80) 80 60 20+100y 8000 40 20 (0,0) 300 (400, 0) 200 100 400

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Tagged in

Math

Algebra

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