A rectangular box with a cover is to have a square base. The volume is to be 10 cubic cm. The surface area of the box in terms of the side x is given by which one of the following function.

Options

A. A. `f(x) = (40//x) + 2x^(2)`
B. B. `f(x)= (40//x) + x^(2)`
C. C. `f(x) = (40//x) + x^(2)`
D. D. `f(x) = (60//x) + 2x`

Answer

To find the surface area of a rectangular box with a square base and a given volume, we can follow these steps:

### Step 1: Define the Variables
Let the side length of the square base be \( x \) cm and the height of the box be \( y \) cm.

### Step 2: Write the Volume Equation
The volume \( V \) of the box is given by the formula:
\[ V = \text{Base Area} \times \text{Height} = x^2 \cdot y \]
Given that the volume is 10 cubic cm, we can write:
\[ x^2 \cdot y = 10 \]

### Step 3: Solve for Height \( y \)
From the volume equation, we can express \( y \) in terms of \( x \):
\[ y = \frac{10}{x^2} \]

### Step 4: Write the Surface Area Equation
The surface area \( S \) of the box with a square base and a cover is given by:
\[ S = 2 \times (\text{Area of Base}) + \text{Area of Sides} \]
The area of the base is \( x^2 \) and there are two bases, so:
\[ \text{Area of Bases} = 2x^2 \]
The area of the sides consists of four rectangles, each with area \( x \cdot y \):
\[ \text{Area of Sides} = 4 \cdot (x \cdot y) = 4xy \]
Thus, the total surface area is:
\[ S = 2x^2 + 4xy \]

### Step 5: Substitute for \( y \)
Now, substitute \( y \) from Step 3 into the surface area equation:
\[ S = 2x^2 + 4x \left(\frac{10}{x^2}\right) \]
This simplifies to:
\[ S = 2x^2 + \frac{40}{x} \]

### Step 6: Final Expression for Surface Area
Thus, the surface area \( S \) in terms of \( x \) is:
\[ S = 2x^2 + \frac{40}{x} \]

### Conclusion
The surface area of the box in terms of the side \( x \) is given by the function:
\[ S(x) = 2x^2 + \frac{40}{x} \]