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Answer:
The maximum volume is 1417.87
Explanation:
<u>Optimization Using Derivatives</u>
We have a 24x30 inch piece of metal and we need to make a rectangular box by cutting a square from each corner of the piece and bending up the sides. The width of the piece is 24 inches and its length is 30 inches
When we cut a square of each corner of side x, the base of the box (after bending up the sides) will be (24-2x) and (30-2x), width and length respectively. The volume of the box is
Operating
To find the maximum value of V, we compute the first derivative and equate it to zero
Simplifying by 12
Completing squares
We have two values for x
The first value is not feasible because it will produce a negative width (24-2(13.58))=-6.16
We'll keep only the solution
The width is
The length is
And the height
The maximum volume is