Answer:
When dimension of box is 33.33 inches × 33.33 inches ×8.33 then its volume is maximum and is 9259.26 cubic inches.
Step-by-step explanation:
Let h be the length (in inches) of the square corners that has been cut out from the cardboard and that would be the height of the cardboard box.
Since the squares have been cut from cardboard, both sides of the cardboard would reduce by 2h.
Thus, The dimension of box is (50 – 2h) × (50 – 2h) × h in dimensions.
The volume V of rectangular box = (Length × Breadth × Height) cubic inches.
..............(1)
Using
For obtaining a box of maximum volume, maximize V as a function of h.
Differentiate both sides with respect to h,
Solving quadratic equation,
For maximum,
thus,
⇒ or
Now check (1) for and
.
is not possible as when h is 25 inches then length and breadth becomes 0.
When .
(1) ⇒
This is the maximum volume the box can assume.
Thus, when dimension of box is 33.3 inches × 33.3 inches ×8.3 then its volume is maximum and is 9259.26 cubic inches.