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 a function that models the surface area \( S \) of a rectangular box with a square base and a volume of \( 40 \, \text{ft}^3 \), we start with the volume formula. Let the side length of the square base be \( x \) and the height be \( h \). The volume \( V \) can be expressed as: \[ V = x^2 h \] Given that \( V = 40 \, \text{ft}^3 \), we have: \[ x^2 h = 40 \implies h = \frac{40}{x^2} \] Next, we calculate the surface area \( S \), which includes the area of the base, the top, and the four sides. The surface area can be expressed as: \[ S = x^2 + x^2 + 4(xh) = 2x^2 + 4(x \cdot h) \] Substituting \( h \) into the surface area equation gives: \[ S = 2x^2 + 4\left(x \cdot \frac{40}{x^2}\right) = 2x^2 + \frac{160}{x} \] So the surface area function in terms of \( x \) is: \[ S(x) = 2x^2 + \frac{160}{x} \] Now you have a neat formula for the surface area based on the length of a side of the base!