Answer:
The dimensions of the box so that total costs are minimum are a side length of 2 feet and a height of 5 feet.
Step-by-step explanation:
Geometrically speaking, the volume of the rectangular box ([tex]V[/tex]), in cubic feet, is represented by this formula:
[tex]V = l^{2}\cdot h[/tex] (1)
Where:
[tex]l[/tex] - Side length of the box, in feet.
[tex]h[/tex] - Height of the box, in feet.
In addition, the total cost of the box ([tex]C[/tex]), in monetary units, is defined by this formula:
[tex]C = (c_{b}+c_{t})\cdot l^{2} + 4\cdot c_{s}\cdot l\cdot h[/tex] (2)
Where:
[tex]c_{b}[/tex] - Unit cost of the base of the box, in monetary units per square foot.
[tex]c_{t}[/tex] - Unit cost of the top of the box, in monetary units per square foot.
[tex]c_{s}[/tex] - Unit cost of the side of the box, in monetary units per square foot.
By (1), we clear [tex]h[/tex] into the expression:
[tex]h = \frac{V}{l^{2}}[/tex]
And we expand (2) and simplify the resulting expression:
[tex]C = (c_{b}+c_{t})\cdot l^{2}+4\cdot c_{s}\cdot \left(\frac{V}{l} \right)[/tex] (3)
If we know that [tex]c_{b} = 0.34\,\frac{m.u.}{ft^{2}}[/tex], [tex]c_{s} = 0.05\,\frac{m.u.}{ft^{2}}[/tex], [tex]c_{t} = 0.16\,\frac{m.u.}{ft^{2}}[/tex] and [tex]V = 40\,ft^{3}[/tex], then we have the resulting expression and find the critical values associated with the side length of the base:
[tex]C = 0.5\cdot l^{2} + \frac{8}{l}[/tex]
The first and second derivatives of this expression are, respectively:
[tex]C' = l -\frac{8}{l^{2}}[/tex] (4)
[tex]C'' = 1 + \frac{16}{l^{3}}[/tex] (5)
After equalizing (4) to zero, we solve for [tex]l[/tex]: (First Derivative Test)
[tex]l-\frac{8}{l^{2}} = 0[/tex]
[tex]l^{3}-8 = 0[/tex]
[tex]l = 2\,ft[/tex]
Then, we evaluate (5) at the value calculated above: (Second Derivative Test)
[tex]C'' = 3[/tex]
Which means that critical value is associated with minimum possible total costs. By (1) we have the height of the box:
[tex]h = 5\,ft[/tex]
The dimensions of the box so that total costs are minimum are a side length of 2 feet and a height of 5 feet.
