construction cost

Question

a carpenter has been asked to build an open box with a square base. the sides of the box will cost $3 per square meter, and the base will cost $4 per square meter. what are the dimensions of the box of greatest volume that can be constructed for $48?

Answer 1

Let x = length of a side of the base

 

     h = height

 

Then, 4(area of base) + 3(area of sides) = 48

 

     So, 4x² + 3(4xh) = 48      12xh = 48 - 4x²

 

                                              h = 4/x - x/3

 

Maximize volume = V = x²h = x²(4/x - x/3)

 

                                    V = 4x - (1/3)x³ , where 0 < x < √12

 

To find the maximum of V, set V' = 0 to locate critical points.

 

   if V' = 0, then 4 - x² = 0

 

                            x = 2 or -2

 

                           Since x can't be negative, x = 2

 

When 0 < x < 2, V' > 0.  So, V is increasing when 0 < x < 2

 

When 2 < x < √12, V' < 0.  So, V is decreasing on 2 < x < √12

 

The volume is maximized  when x = 2 meters

 

                                             h = 4/x - x/3 = 2 - 2/3 = 1 1/3 meters