Question

an open rectangular box (no top) with volume 6 cubic meters has a square base. express the surface area of the box as a function of the length of one of the sides of the base. ( ) =

Answer 1

The surface area of an open rectangular box with a square base as a function of the side length s is SA(s) = s² + 24/s. This is derived by first solving for the box's height in terms of its volume and base side length, then calculating the surface area, considering the base and the four side faces.

Step-by-step explanation:

To express the surface area of an open rectangular box with a square base as a function of the length of one of the sides of the base, we start by letting the side of the square base be s. Since the base is square, the area of the base is s². The volume of the box, which is 6 cubic meters, is equal to the area of the base times the height h (V = s² × h).

First, we express the height of the box as a function of s:

1. V = s² × h
2. 6 = s² × h
3. h = 6 / s²

Next, we calculate the surface area (SA) of the box, noting that the box has no top. The box has one square base and four sides. Therefore, the SA consists of four side faces and one base:

1. SA = Base area + 4 × (Side area)
2. SA = s² + 4 × (s × h)
3. Substitute h from step 3 into the SA equation
4. SA = s² + 4 × (s × (6 / s²))
5. SA = s² + 24 / s

Therefore, the surface area of the box as a function of the side length s is SA(s) = s² + 24 / s.