Answer:
x = 7.45 inch for the maximum volume.
The area of the square = x² = 7.45² = 55.5025 inch²
Step-by-step explanation:
From the given information:
An open-top box is to be made from a 42-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side.
The objective is to determine the length of the side x of the square cut out of each corner to get a box with the maximum volume
The volume of the box = l×b×h
The volume of the box =
The volume of the box =
The volume of the box =
The volume of the box =
The volume of the box =
For the maximum volume V' = 0
V' =
Using the quadratic formula; we have:
where;
a = 12 , b = -360 c = 2016
For the maximum value , we check the points in the second derivative term
V'' = 24x - 360
V'' ( 22.55) = 24(22.55) - 360
V'' ( 22.55) = 541.2 - 360
V'' ( 22.55) = 181.2 (minimum)
V'' ( 7.45) = 24(7.45) - 360
V'' ( 7.45) = 178.8 - 360
V'' ( 7.45) = -181.2 < 0 (maximum)
Therefore, x = 7.45 inch for the maximum volume.
The area of the square = x² = 7.45² = 55.5025 inch²