negative answer doesn’t make any sense in the case, so we use the solution x = 2. We can then
find the area by plugging in x = 2 to get A = 24(2) − 2(2)^3 = 32. We can verify that this is a
maximum by taking the second derivative: = −12x. Next, we plug in x = 2: =
−12(2) = −24. Because the value of the second derivative is negative, according to the second
derivative test (see this page), the area is a maximum at x = 2.
- x = ≈ 1.697 inches
First, let’s draw a picture.
After we cut out the squares of side x and fold up the sides, the dimensions of the box will be:
width: 9 − 2x; length: 12 − 2x; depth: x.
Using the formula for the volume of a rectangular prism, we can get an equation for the volume
of the box in terms of x: V = x(9 − 2x)(12 − 2x) = 108x − 42x^2 + 4x^3.
Now, we take the derivative: = 108 − 84x + 12x^2 . Next, we set the derivative equal to
zero: 108 − 84x + 12x^2 = 0. If we solve this for x, we get x = ≈ 5.303, 1.697. We
can’t cut two squares of length 5.303 inches from a side of length 9 inches, so we can get rid of
that answer. Therefore, the answer must be x = ≈ 1.697 inches. We can verify that
