An open box is to be made from a rectangular sheet of cardboard od dimension 16cm by 24cm by cutting out squares of = size from each of the four corners & bending up flaps.Find the dimensions of box of largest volume that can be made?
1 Answer
17,72 cm x 9,722 cm x 3,139 cm
Explanation:
I first drew a sketch of the entire sheet showing the corners of dimensions x being cut out, and then showed the box with its dimensions in terms of x.
I then found the volume of the box in terms of x.
I then differentiate the volume function and set it to zero as this is the condition for finding relative minimum and maximum values of a function.
Since the derivative is a 2nd degree polynomial, I used the quadratic formula to find its roots. The one root is impossible as it implies the whole box length being cut away, so I discard it.
The other root I check using the 2nd derivative test and find that it outputs a negative value in the second derivative of the volume function, so according to the theory it is a relative maximum.
I then substitute the value into the box dimensions to give the dimensions which would yield the maximum volume.
Full details in sketch.
