Calc derivatives - Minimum total surface area in a box of V = 160 ft2

• Dave_1420
In summary, the conversation discusses finding the dimensions of a box with a square base and open top that has a volume of 160 cubic feet. The total surface area of the box is to be minimized using calculus. The conversation also includes a question about how to formulate an expression for the total outer surface area of the box using the dimensions of the base and height.
Dave_1420
Hi everyone! I'm new to online math forums. I wonder if anyone can give me a hand on this - it would be greatly appreciated.

Thank you in advance!

Dave

Homework Statement

If a box with a square base and an open top is to have a volume of 160 cubic feet, find the dimensions of the box having the minimum total surface area. Use calculus to find the solution.

Homework Equations

If X are the sides of the base, and Y is the height of the box, what would be the formula to find surface? How can the minimum surface be calculated using derivative applications?

The Attempt at a Solution

Surface = minimum possible

In terms of x and y, formulate an expression for the total outer surface area of the box.

If "x" is the length of a side of the square base, y is the height of the box, then the base is an "x by x" square and there are four sides that are "x by y" rectangles. Do you know how to find the area of squares and rectangles?

1. What is the formula for finding the minimum total surface area in a box with a volume of 160 ft2?

The formula for finding the minimum total surface area in a box with a volume of 160 ft2 is A = 2xy + 2xz + 2yz, where x, y, and z are the dimensions of the box.

2. How do you calculate the dimensions of the box with the minimum total surface area?

To calculate the dimensions of the box with the minimum total surface area, you can use the derivative of the formula for surface area with respect to one of the variables (x, y, or z), set it equal to 0, and solve for the variable. This will give you the value of the variable that minimizes the surface area.

3. Can the minimum total surface area be negative?

No, the minimum total surface area cannot be negative. Surface area is a measure of the outside of a three-dimensional object, and therefore it cannot be negative.

4. What units should be used when solving for the dimensions of the box?

The units used for solving for the dimensions of the box will depend on the units used for the volume. For example, if the volume is given in cubic feet, the dimensions will also be in feet.

5. Are there any real-life applications for this concept?

Yes, this concept can be applied to real-life situations, such as designing packaging for products or optimizing the use of space in a room. It can also be used in engineering and architecture to determine the most efficient design for a structure.

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