Maximize Volume of a Box
How to maximize the volume of a box using the first derivative of the volume.
Problem 1: A sheet of metal 12 inches by 10 inches is to be used to make a open box. Squares of equal sides x are cut out of each corner then the sides are folded to make the box. Find the value of x that makes the volume maximum.
Solution to Problem 1:

We first use the formula of the volume of a rectangular box.
V = L * W * H

The box to be made has the following dimensions:
L = 12  x
W = 10  2x
H = x

We now write the volume of the box to ba made as follows:
V(x) = x (12  2x) (10  2x) = 4x (6  x) (5  x)
= 4x (x^{ 2} 11 x + 30)

We now determine the domain of function V(x). All dimemsions of the box must be positive or zero, hence the conditions
x > = 0 and 6  x > = 0 and 5  x > = 0

Solve the above inequalities and find the intersection, hence the domain of function V(x)
0 < = x < = 5

Let us now find the first derivative of V(x) using its last expression.
dV / dx = 4 [ (x^{ 2} 11 x + 3) + x (2x  11) ]
= 3 x^{ 2} 22 x + 30

Let us now find all values of x that makes dV / dx = 0 by solving the quadratic equation
3 x^{ 2} 22 x + 30 = 0

Two values make dV / dx = 0: x = 5.52 and x = 1.81, rounded to one decimal place. x = 5.52 is outside the domain and is therefore rejected. Let us now examine the values of V(x) at x = 1.81 and the endpoints of the domain.
V(0) = 0 , v(5) = 0 and V(1.81) = 96.77 (rounded to two decimal places)

So V(x) is maximum for x = 1.81 inches. The graph of function V(x) is shown below and we can clearly see that there is a maximum very close to 1.8.
More references on
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