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An engineer is designing an open (without top) rectangular prism with a square base. The box’s volume must be ${1800\ {\rm in^{3}}}$ . The engineer wants to minimize the box’s surface area.

There is a rectangular prism. Its base is a square with its side length marked as x in. The height of the prism is marked as h in.

Let the square base’s side length be $x$ in, and the prism’s height be $h$ in.

The box has 5 sides, and its surface area is $4xh+x^2$ .

It’s given that the box’s volume is ${1800\ {\rm in^{3}}}$ , we have:

$\displaystyle{\begin{aligned} \text{volume} &= (\text{base area})(\text{height}) \\ 1800 &= x^2h \\ \frac{1800}{x^2} &= h \end{aligned} }$

Now we can write the box’s surface area, $S(x)$ , as a function of $x$ :

$\displaystyle{\begin{aligned} S(x) &= 4xh+x^2 \\ S(x) &= 4x\left(\frac{1800}{x^2}\right)+x^2 \\ S(x) &= \frac{7200}{x}+x^2 \\ \end{aligned} }$

Use graphing technology to find the value of $x$ and $h$ such that the box has a minimum surface area. Round your answers to two decimal places.

The box has a minimum surface area of when its base’s side length is and the box’s height is .

(Use in^2 for square inches.)

You can earn partial credit on this problem.