Cracking The Ap Calculus ab Exam 2018

After we cut out the squares of side x and fold up the sides, the dimensions of the box will be

width:  18  −   2x

length: 24  −   2x

depth:  x

Using the formula for the volume of a rectangular prism, we can get an equation for the volume in terms of
x.

V   =   x(18    −   2x)(24  −   2x)

Multiply the terms together (and be careful with your algebra).

V   =   x(18    −   2x)(24  −   2x) =   4x^3    −   84x^2   +   432x

Now take the derivative.

    =   12x^2   −   16x +   432

Set the derivative equal to zero, and solve for x.

12 x^2  −   168x    +   432 =   0

x^2     −   14x +   36  =   0

x   =       =   7   ±       ≈   3.4,    10.6

Common sense tells us that you can’t cut out two square pieces that measure 10.6 inches to a side (the
sheet’s only 18 inches wide!), so the maximizing value has to be 3.4 inches. Here’s the second derivative
test, just to be sure.