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 (
), in cubic feet, is represented by this formula:
(1)
Where:
- Side length of the box, in feet.
- Height of the box, in feet.
In addition, the total cost of the box (
), in monetary units, is defined by this formula:
(2)
Where:
- Unit cost of the base of the box, in monetary units per square foot.
- Unit cost of the top of the box, in monetary units per square foot.
- Unit cost of the side of the box, in monetary units per square foot.
By (1), we clear
into the expression:
![h = \frac{V}{l^{2}}](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7BV%7D%7Bl%5E%7B2%7D%7D)
And we expand (2) and simplify the resulting expression:
(3)
If we know that
,
,
and
, then we have the resulting expression and find the critical values associated with the side length of the base:
The first and second derivatives of this expression are, respectively:
(4)
(5)
After equalizing (4) to zero, we solve for
: (First Derivative Test)
![l-\frac{8}{l^{2}} = 0](https://tex.z-dn.net/?f=l-%5Cfrac%7B8%7D%7Bl%5E%7B2%7D%7D%20%3D%200)
![l^{3}-8 = 0](https://tex.z-dn.net/?f=l%5E%7B3%7D-8%20%3D%200)
![l = 2\,ft](https://tex.z-dn.net/?f=l%20%3D%202%5C%2Cft)
Then, we evaluate (5) at the value calculated above: (Second Derivative Test)
![C'' = 3](https://tex.z-dn.net/?f=C%27%27%20%3D%203)
Which means that critical value is associated with minimum possible total costs. By (1) we have the height of the box:
![h = 5\,ft](https://tex.z-dn.net/?f=h%20%3D%205%5C%2Cft)
The dimensions of the box so that total costs are minimum are a side length of 2 feet and a height of 5 feet.