How do you determine the surface area of an open box with a square base that has a volume of 10 cubic feet, expressed as a function of the base’s side length x?

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Determining the Surface Area of an Open Box

An open box with a square base means that it has four vertical sides and one top, but no bottom. To find the surface area as a function of the side length of the base, denoted as x, we’ll follow a few steps to derive the formula.

Step 1: Understand the Volume Formula

The volume (V) of a box is calculated using the formula:

 V = length × width × height

For our open box with a square base:

 V = x × x × h = x²h

Given that the volume is 10 cubic feet, we can set up the equation:

 x²h = 10

From this, we can solve for the height (h):

 h = 10 / x²

Step 2: Determine the Surface Area

The surface area (S) of an open box can be calculated using the formula that sums the area of the base and the areas of the four sides:

 S = Area of base + Area of sides

Since we have a square base:

 Area of base = x²

Each side of the box has a height of h, so the area of the four sides is:

 Area of sides = 4 * (x * h) = 4xh

Combining these, we can express the total surface area S as:

 S = x² + 4xh

Step 3: Substitute for h

Now, substituting our expression for h from earlier:

 S = x² + 4x(10 / x²)

Simplifying this, we find:

 S = x² + (40 / x)

Therefore, the surface area S as a function of the base side length x is given by:

 S(x) = x² + 40/x

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