Question

From a 24-inch by 24-inch piece of metal, squares are cut out of the four corners so that the sides can then be folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are

cut out. Express the volume of the box as a function of × Graph the function and from the graph determine the value of x, to the nearest tenth of an inch, that will yield the maximum volume

Answer 1

Explanation:

To find the volume of the box after cutting out squares of length x from each corner, we need to consider the dimensions of the resulting box. After cutting out squares of length x from each corner, the dimensions of the box will be (24-2x) inches by (24-2x) inches, with a height of x inches.

The volume (V) of the box can be expressed as:

V = (24 - 2x) * (24 - 2x) * x

Now, let's simplify the equation:

V = (576 - 48x - 48x + 4x^2) * x

V = (4x^2 - 96x + 576) * x

V = 4x^3 - 96x^2 + 576x

To find the value of x that yields the maximum volume, we need to find the critical points. To do this, we take the derivative of the volume function with respect to x and set it equal to zero:

dV/dx = 12x^2 - 192x + 576

Now, set dV/dx = 0:

12x^2 - 192x + 576 = 0

Divide the equation by 12 to simplify:

x^2 - 16x + 48 = 0

Now, we can use the quadratic formula to solve for x:

x = (-(-16) ± √((-16)^2 - 4*1*48)) / (2*1)

x = (16 ± √(256 - 192)) / 2

x = (16 ± √64) / 2

x = (16 ± 8) / 2

Now, we have two possible values for x:

1. x = (16 + 8) / 2 = 24 / 2 = 12 inches

2. x = (16 - 8) / 2 = 8 / 2 = 4 inches

We discard the negative value of x since it doesn't make sense in the context of the problem (we can't have negative side lengths). So, the valid value for x that yields the maximum volume is approximately 12 inches.

To graph the function, we can plot the volume (V) as a function of x:

V(x) = 4x^3 - 96x^2 + 576x

The x-axis represents the value of x (side length of the squares cut out), and the y-axis represents the volume of the box. The maximum point on the graph corresponds to the value of x that gives the maximum volume, which we have already determined to be approximately 12 inches.