Question

Find a formula for the described function. An open rectangular box with volume $4{\mathrm{m}}^3$ has a square base. Express the surface area $\mathrm{SA}$ of the box as a function of the length of a side of the base, $x$. $\mathrm{SA}=$${\mathrm{m}}^2$ State the domain of $\mathrm{SA}$. (Enter your answer in interval notation).

Answer 1

Finding the surface area and domain of $\mathrm{SA}$:

Given an open rectangular box with volume $4{\mathrm{m}}^3$ has a square base.

The length of a side of the base is $x$.
Volume $=x·x·h$
$\Rightarrow x^{2}h=4$
$\Rightarrow h=\frac{4}{x^2}$
Surface area can be found as :
$x^2+4hx=x^2+(4x\times \frac{4}{x^2})$
Surface area $=x^2+\frac{16}{x}{\mathrm{m}}^2$
Any domain will be $x\in (0,\infty )$ we can also say $x>0$
$x$ is greater than zero as the surface area will always be a positive value and $x$ cannot be negative.

Hence, the surface area and domain is $x^2+\frac{16}{x}{\mathrm{m}}^2$ and $x\in (0,\infty )$ respectively.