What are the constraints for this problem?

Question

Webster Instruments is opening a new music shop in which they sell guitars and basses. The store is small, so they can buy no more than 50 instruments total. The store’s cost to buy a guitar in $200 and $400 for the basses. They have $12000 to spend on buying their initial inventory. because guitars are more popular than basses, the numbers of guitars must be at least twice the number of basses. Their supplier requires a minimum initial purchase of at least 17 guitars and at least 5 basses. If each bass will sell for $900 and each guitar sell for $750, determine how many of each instrument will maximize the profit. What is the maximum profit? How much will they spend in purchasing their initial inventory?

Answer 1

Whew, this is a busy one. Let's work through it. Let x = the number of guitars and y = the number of basses.

1. they can buy no more than 50 instruments total: x + y ≤ 50
2. The store’s cost to buy a guitar is $200 and $400 for the basses so the Cost = 200x + 400y. They have $12000 to spend on buying their initial inventory, so 200x + 400y ≤ 12,000
3. because guitars are more popular than basses, the numbers of guitars must be at least twice the number of basses: x ≥ 2y
4. Their supplier requires a minimum initial purchase of at least 17 guitars and at least 5 basses: x ≥ 17 and y ≥ 5
5. each bass will sell for $900 and each guitar sell for $750, so Revenue = 750x + 900y. The Profit = Revenue - Cost = 750x + 900y - (200x + 400y) = 550x + 500y. This is the equation to be maximized by applying linear programming given the above constraints.

Answer 2

I tried to create a video answer for this question, but WyzAnt went into an endless loop. So here are the constraints along with the profit function. Also at the end is a link to the Desmos graph that was referenced in the video attempt. Let x = #guitars, y = #basses.

x+y≤50

200x+400y≤12000

x ≥ 2y

x ≥ 17

y ≥ 5

You will have to find the points of intersection of all the borders of those regions and feed the coordinates into the "Profit" function:

P(x,y) = 750x + 900y

Hopefully you are aware of the URL to the main Desmos Graphing calculator and can append the following to that URL to visit the referenced Desmos graph.

calculator/beimo56wpg

Answer 3

g+b=50

2xg200+xb400=1200 at least.

I'd purchase the minimum "b"s; the rest "g"s.

With only $1200; there's nothing you can do with the requirements.