Consider a 10in by 14in rectangle to which you make square cuts of side length "x" in each corner, then fold the sides and form a box of height "x". What should "x" be to maximize the volume of the box?
Calculus hmm.
Well, we know a box has the formula:
V=(L)(W)(H).
The box is 10 by 14 so:
L = 10-2x and W = 14-2x H = x
V(x) = x(10-2x)(14-2x)
Simplifying this...
V(x) = 4x^3-48x^2+140x
The volume has to be greater than 0, of course, so:
0<x<5, or else the shape wont exist!
Find the first derivative:
dV/dx (4x^3-48x^2+140x) - power rule.
12x^2-96x+140=0.
Solve using the quadratic equation...
The zeroes are 1.92 and 6.08; 6.08 is a relative minimum, and outside the interval 0<x<5, therefore the answer is:
x= 1.92