I'm not sure how to write the total cost of building a box in terms of one the lengths of the base.

Question

The question srates:

A square base box has a volume of 500 ft^3. The cost of the material for the top and the bottom (square base) of the box is $18 per ft^2. The cost for both sides is $3 per ft^2. Express the total cost of this box in terms of one the lengthe of base. 

Answer 1

Let us assume x ft is the lenght of base and y is the height of the box.

 

The volume of the box is x²y =500ft^3 

 

i.e y=500/x²                                            (1)

 

Now let calculate the cost of the materials for the building.The top and bottom sides are squares with size x.The 4 sides are rectangles with breadth x and height y

 

So the cost C=18*2*x²+3*4*x*y=36x²+12xy            (2)

Subtitute (1) in (2) to eliminate y

 

C=36x2+12x* 500/x² 

 

C=36x²+6000/x $

 

You can minimize C to determine the value of x for cost .

 

Thank you