ASK YOUR TEACHER \( \begin{array}{l}\text { A rectangular box with a volume of } 40 \mathrm{ft}^{3} \text { has a square base. Find a function that models its surface area } S \text { in terms of the length } \\ x \text { of one side of its base. } \\ S(x)=\square\end{array} \)

To find the surface area \( S \) in terms of the length \( x \) of one side of the base of the rectangular box, we start by noting the dimensions of the box. Since the base is square and has side length \( x \), the area of the base is \( x^2 \). Let \( h \) be the height of the box. The volume \( V \) of the box is given by: \[ V = \text{Base Area} \times \text{Height} = x^2 h \] Given that the volume is \( 40 \, \text{ft}^3 \), we can set up the equation: \[ x^2 h = 40 \implies h = \frac{40}{x^2} \] Next, let's calculate the surface area \( S \). The surface area includes the area of the base, the top (which is also \( x^2 \)), and four sides (each side has area \( xh \)): \[ S = \text{Area of Base} + \text{Area of Top} + \text{Area of Sides} \] \[ S = x^2 + x^2 + 4(xh) = 2x^2 + 4(xh) \] Substituting \( h \) into the surface area equation: \[ S = 2x^2 + 4\left(x \cdot \frac{40}{x^2}\right) \] \[ S = 2x^2 + \frac{160}{x} \] Therefore, the function that models the surface area in terms of the side length \( x \) of the base is: \[ S(x) = 2x^2 + \frac{160}{x} \]