Mendoza Manufacturing plans to producea one compartment vertical file by bending the long side of a 10-in. by 18-in. sheet of plastic along two lines to form a U-shape. How tall should the file be in order to maximize the volume that it can hold?

The width is 10 in.

The long side is 18 in.
Let x be the distance of the bend from the edge, then the volume is
V(x) = 10x(18-2x)
Differentiate V(x) with respect to x to get V'(x).
Solve for x in V'(x) = 0 (solution = h)
Check that V"(h) (second derivative) is negative for a maximum.
If you have not done differential calculus before, please post again.

Well, if we're talking about file storage, it's important to keep things upright and organized. As for the height of the file, let's see if the math can give us a laugh.

To maximize the volume, we need to find the height that gives us the biggest U-shaped file. So, let's get down to business!

First, let's assume the height of the file is 'h'. The width of the file is 10 inches, and since we're folding it along two lines, the final width will be h inches.

Now, let's take a closer look at the length. The original length is 18 inches, but once the sheet is folded into a U-shape, we have to subtract twice the height from the length.

So, the length of the file will be 18 - 2h inches.

Now that we have the dimensions, we can calculate the volume of our U-shaped file using the formula:

Volume = length × width × height

Volume = (18 - 2h) × 10 × h

And now, the grand finale! To maximize the volume, we need to find the value of 'h' that gives us the maximum. So, let calculus come in and do its magic – find the derivative of the volume equation and set it equal to zero.

However, I'm not going to leave you hanging (or filing) without a punchline. So here it is: "Why did the file go to the comedy club? Because it wanted to be a stand-up file!"

Alright, back to the math. Solve the derivative equation, and you'll find the value of 'h' that maximizes the volume. Congratulations, you've cracked the case of the ultimate file height!

To maximize the volume, we need to determine the height of the file. Let's break down the problem step-by-step:

1. Determine the dimensions of the U-shaped file:
- The original sheet of plastic is 10 inches by 18 inches.
- When bent, the width of the file becomes the height, and the length becomes the base.

2. Define the variables:
- Let h represent the height (width) of the file.
- Let x represent the length (base) of the file.

3. Set up the equation for the volume:
- The volume V of the file is given by V = base × width × height.
- Since the base of the file is x and the width (height) is h, the equation can be written as V = x × h × h.

4. Express the base in terms of the height:
- The sheet is bent along two lines, so the base has two lengths. The total length is equal to the sum of the two lengths of the sides of the file, which is x + x = 2x.

5. Substitute the expression for the base into the volume equation:
- V = (2x) × h × h = 2xh².

6. Determine the equation for the volume in terms of one variable:
- Since we want to maximize the volume, we need the equation in terms of a single variable. In this case, it will be in terms of h.
- Rewrite the equation as V(h) = 2xh², where x is a constant and represents the base length.

7. Maximize the volume by finding the derivative:
- Take the derivative of V(h) with respect to h, which will yield V'(h) = 4xh.

8. Set the derivative equal to zero and solve for h:
- Set V'(h) = 0 and solve for h.
- 4xh = 0
- h = 0 / 4x
- h = 0

9. Evaluate the second derivative to determine the nature of the critical point:
- Take the second derivative of V(h) with respect to h.
- V''(h) = 4x, which is a constant.

10. Analyze the critical point:
- Since the second derivative is always positive, the critical point at h = 0 is a minimum. However, a file with zero height is not possible, so we can disregard this solution.

11. Consider the boundaries:
- The height cannot be greater than 10 inches or less than zero (as it represents the width).
- Therefore, the only possible solution within the given constraints is h = 10 inches.

12. Determine the maximum volume:
- Calculate the volume using the value of h = 10 inches and the original base length x.
- V = 2xh² = 2x(10)² = 200x.

Therefore, to maximize the volume of the file, the height (width) of the file should be 10 inches.

To find the height that will maximize the volume of the one compartment vertical file, we can use the process of optimization.

Step 1: Define the problem
We want to maximize the volume of the file. Let's call the height of the file "h".

Step 2: Formulate the objective function
The volume of a rectangular prism is given by V = lwh, where l is the length, w is the width, and h is the height. In this case, the length is fixed at 18 inches, the width is fixed at 10 inches, and we want to find the value of h that maximizes the volume.

Step 3: Express the constraint
In this problem, there are no explicit constraints given, but we need to consider the physical constraint that the height cannot exceed the length of the sheet, which is 18 inches. So the constraint is h ≤ 18.

Step 4: Solve the problem
Since we want to maximize the volume, we need to find the critical points of the volume function. To do this, we can take the derivative of the volume function with respect to h and set it equal to zero.

V = lwh
V = 18wh
dV/dh = 18w

Setting dV/dh = 0:
18w = 0

Solving for w, we find that w = 0.

This means that the derivative is zero when w equals zero. However, since we need to have a non-zero width for the file to be meaningful, we can discard this solution.

So, there are no critical points for the volume function.

Next, we need to check the endpoints of the feasible region, which in this case is the maximum height, h ≤ 18.

When h = 18, the volume is given by:
V = 18 * 10 * 18
V = 3240 cubic inches

Therefore, at h = 18, the volume is 3240 cubic inches.

Since there are no critical points and the volume is continuous on the feasible region, we can conclude that the maximum volume is achieved when the height of the file is 18 inches.