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Step-by-step explanation:
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:
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)
Then, we evaluate (5) at the value calculated above: (Second Derivative Test)
Which means that critical value is associated with minimum possible total costs. By (1) we have the height of the box:
The dimensions of the box so that total costs are minimum are a side length of 2 feet and a height of 5 feet.