This is one neither of us could get because it had a constraint that was greather than or equal to. If you would deign to help me, I would greatly appreciate it.
Pizza shop makes 12 and 16 inch pizzas. 12 inch $2 profit. 16 inch $4 profit. 12 inch takes 1/5 hour to prepare, 16 inch 1/4 hour to prepare. Workers put in at most 240 hours per week. They must prepare at least 1000 boxes to meet demands. What number of each type pizza will produce the greatest profit for the company.
When we graph the constraints we can’t end up with a polygon because of the statement “at least” 1000.
Any help will be appreciated.
200 x 12" -> $400, 40 hours
800 x 16" -> $3,200, 200 hours
Didn’t mean to hit submit yet.
Anyway, you just need to recognise that if you make n 12" pizzas, you can only make 4(240 - n/5) 16" pizzas.
Or the long road…
x = 12" pizzas made
y = 16" pizzas made
For reference, I’ll label each formula as #a, #b, etc…
#a: x/5 + y/4 = 240
#b: x + y >= 1000
Maximize #c: 2x + 4y
#a implies
#d: y = (4800 - 4x)/5
combining #b and #d:
(5x + 4800 - 4x)/5 >= 1000
#e: x >= 200
combining #c and #d:
We want to maximize (10x + 4*4800 - 16x)/5 =
#f: (19200 - 6x)/5
We see in #f that this is a straight line that slopes down. If we work it out, we see that at x=0, profit=3840, and at x=3200, profit=0.
So, since the profit slopes down and #e says x>=200, we know that x actually equals 200. (a higher value would be a lower profit.
Plugging x in gives us a y of 800
So…
x=200 (for 40 hours)
y=800 (for 200 hours)
Profit = $19,200