To find the surface area of the open box as a function of the height x, we start by expressing both the volume and surface area in terms of x.
1. **Volume**: The volume V of an open box with a square base can be calculated using the formula:
V = x * (x^2) = x^3
Here, x is the height and x2 is the area of the base. We want this volume to be 10 cubic feet. Therefore, we set:
x^3 = 10
2. **Solving for x**: This gives us:
x = (10)^(1/3)
So, x ≈ 2.154 feet (this value is for height).
3. **Surface Area**: The surface area S of the box can be calculated using the formula:
S = x^2 + 4xh
Where:
– x^2 is the area of the base,
– 4xh is the area of the four sides (since it’s an open box, we don’t include a top).
Substituting h = 10/x^2 (from the volume equation) into the surface area formula, we get:
S = x^2 + 4x(10/x^2) = x^2 + 40/x
Thus, the surface area as a function of x is:
S(x) = x^2 + 40/x
This equation allows you to calculate the surface area based on the height x of the box.