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 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 ot an inch, that will yield the maximum volume x can be 3.6,3.8,4.2,4.0

Answer 1

x = 4.0 inches is the value that will yield the maximum volume to the nearest tenth of an inch.

Explanation:

To express the volume of the box as a function of x, we follow the same steps as before:

1. First, we determine the dimensions of the 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.

2. 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

Now, we can graph the function V(x) = 4x^3 - 96x^2 + 576x to visualize its behavior and find the value of x that yields the maximum volume.

Here is the graph:

(Note: The graph shows the volume V as a function of x for the range of x-values given: x = 3.6, 3.8, 4.0, and 4.2)

Now, to determine the value of x that yields the maximum volume, we need to identify the highest point on the graph. From the graph, it appears that the maximum volume occurs at approximately x ≈ 4.0 inches.

Therefore, from the given options, x = 4.0 inches is the value that will yield the maximum volume to the nearest tenth of an inch.