A company makes three sizes of cardboard boxes: small, medium, and large. It cos

Question

A company makes three sizes of cardboard boxes: small, medium, and large. It costs $2.50 to make a small box, $4.00 for a medium box, and $4.50 for a large box. Fixed costs are $8000.

a) express the cost of making x small boxes, y medium boxes, and z large boxes as a function of three variables: C=f(x,y,z).

b) Find f(3000,5000,4000) and interpret it.

c) what is the domain of f?

There was an answer to this problem on here but I need to know how to get those answers.  I thought a) would be $8000 = 2.50x + 4.00y + 4.50z.  As for b) and c) I'm totally stumped.  This hasn't been covered in class so I have no idea how to go about it.  It seems like it should be very simple. Thanks in advance. :)

Explanation / Answer

8000$ being fixed cost means that this will always factor in the total cost no matter how many small, medium or large boxes are made. (ie irrespective of values of x, y and z, 8000$ will always be added to the cost).

Thus,

a. Cost = f(x,y,z) = 2.5x + 4y + 4.5z + 8000

b. f(3000,5000,4000) = 2.5*3000 + 4*5000 + 4.5*4000 + 8000
= 7500 + 20000 + 18000 + 8000

= 53500$

c. The domain of f(x,y,z) will be x >= 0 , y >= 0 and z >= 0 since the number of boxes cannot be negative but can be zero for a particular type.

( ">=" is "greater than or equal to" )

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