An open box having a square base is to be constructed from 32 square inches of material. What should be the dimensions of the box to obtain a max volume?
To find the maximum volume of the open box, we need to determine the dimensions that will optimize the volume. Let's break down the problem step by step:
Step 1: Define the variables:
Let's assume the side length of the square base to be "x", and the height of the box to be "h".
Step 2: Determine the volume of the box:
The volume (V) of the box can be calculated by multiplying the area of the base (x * x = x^2) by the height (h).
V = x^2 * h
Step 3: Determine the constraint:
The constraint in this problem is that the total surface area of the box should be 32 square inches.
The total surface area (A) of the box can be calculated by adding up the area of the base (x^2) and the combined areas of the four sides (4x * h).
A = x^2 + 4xh
However, we have a constraint: A = 32.
Step 4: Express the height in terms of "x":
Rearrange the constraint equation to solve for h:
h = (32 - x^2) / (4x)
Step 5: Express the volume in terms of "x" only:
Substitute the expression for "h" into the volume equation found in Step 2:
V = x^2 * [(32 - x^2) / (4x)]
Simplify:
V = (8x - x^3) / 4
Step 6: Find the maximum volume:
To find the maximum volume, we need to find the critical points of the volume function. We can do this by taking the derivative of the volume function with respect to "x", setting it equal to zero, and solving for "x".
dV/dx = 8/4 - 3x^2/4
0 = 2 - (3x^2/4)
3x^2/4 = 2
x^2 = 8/3
x = sqrt(8/3)
x ≈ 1.6329
Step 7: Calculate the maximum volume:
Substitute the value of "x" into the volume equation (V = (8x - x^3) / 4) to find the maximum volume.
V = [(8 * 1.6329) - (1.6329^3)] / 4
V ≈ 6.8571 cubic inches
Step 8: Determine the dimensions of the box:
Since we have the value of "x", the side length of the square base is approximately 1.6329 inches. To find the height, substitute the value of "x" into the expression for "h" from Step 4:
h = (32 - x^2) / (4x)
h = (32 - (1.6329^2)) / (4 * 1.6329)
h ≈ 5.1427 inches
Therefore, to obtain the maximum volume, the dimensions of the box should be approximately 1.6329 inches for the side length of the square base and 5.1427 inches for the height.