How many engines should be shipped to each plant to minimize shipping​ costs?

[brieannalarkson2016]

A manufacturing company receives orders for engines from two assembly plants. Plant 1 needs at least 45 engines, and Plant 2 needs at least 32 engines. The company can send at most 140 engines to these assembly plants. It costs $35 per engine to ship to Plant 1 and $50 per engine to ship to Plant 2. Plant 1 gives the manufacturing company $20 in rebates toward its products for each engine they buy, while Plant 2 gives similar $15 rebates. The manufacturer estimates that they need at least $1500 in rebates to cover products they plan to buy from the two plants. How many engines should be shipped to each plant to minimize shipping costs? What is the minimum cost?

The number of engines to send to Plant 1 is: . . . . .

The number of engines to send to Plant 2 is: . . . . .

The minimum shipping cost is: . . . . .

 

[mmm4444bot]

A manufacturing company receives orders for engines from two assembly plants. Plant 1 needs at least 45 engines, and Plant 2 needs at least 32 engines. The company can send at most 140 engines to these assembly plants. It costs $35 per engine to ship to Plant 1 and $50 per engine to ship to Plant 2. Plant 1 gives the manufacturing company $20 in rebates toward its products for each engine they buy, while Plant 2 gives similar $15 rebates. The manufacturer estimates that they need at least $1500 in rebates to cover products they plan to buy from the two plants. How many engines should be shipped to each plant to minimize shipping costs? What is the minimum cost?

A common way to solve this type of problem is by using a method known as Linear Programming. I'm guessing that you're supposed to use this method.

If so, where did you get stuck? What have you done so far?

The first step is to define symbols x and y (one of them needs to represent the number of engines shipped to Plant I, and the other represents the number shipped to Plant II). What did you decide?

You then use these symbols, to write four inequalities, representing the various constraints stated in the exercise.

You will also need an expression (using x and y) to represent the total shipping cost.

What are your thoughts, up to this point?

Later, you'll need to graph the inequalities -- the graphs enclose a four-sided region. Next, you determine the (x,y) coordinates, at each vertex of the region. The last step is to evaluate the shipping-cost expression, trying each pair of coordinates, to discover which pair yields the lowest result.

If you're not familiar with Linear Programming, let us know. We can provide links to lessons and worked examples for you to study.

Please also ensure that you have read the forum guidelines. Thank you. :cool: