I have a 3D rectangular object. It has a volume of $75\text{cm}^3$; it has a square base with one edge of the base having a length of $x$. Find a function $S$, that represents the surface area of the object with respect to $x$.
I know that the surface area $S$ of a 3D rectangular object is a function of its length ($L$), width ($W$), and height ($H$). That is, adding the areas of each face gives the surface area. The area of each face is below:
$$\begin{align}
S_1 &= LW \\
S_2 &= LW \\
S_3 &= WH \\
S_4 &= WH \\
S_5 &= HL \\
S_6 &= HL
\end{align}$$
Or more declarative, the surface area of a 3D rectangular object is:
$$\begin{align}
S &= LW + LW + WH + WH + HL + HL \\
&= 2LW + 2WH + 2HL
\end{align}$$
I also know the volume $V$ of a 3D rectangular object is a function of its length, width, and height:
$$V = LWH$$
Using algebra I can find any given dimension:
$$\begin{align}
L &= V/WH \\
W &= V/LH \\
H &= V/LW
\end{align}$$
Going back to the shape at hand, I know the area of one of the faces of the polygon: $x^2$. This corresponds to the length multiplied by width.
Since I know the volume, length, and width, I can get the height, too:
$$\begin{align}
H &= V/LW \\
&= 75/x^2
\end{align}$$
Substituting $H$ into the equation for the surface area of a 3D rectangular object:
$$\begin{align}
S &= 2LW + 2WH + 2HL \\
&= 2(x^2) + 2(\frac{75}{x^2})(x) + 2(\frac{75}{x^2})(x) \\
&= 2x^2 + 2x(\frac{75}{x^2}) + 2x(\frac{75}{x^2}) \\
&= 2x^2 + 4x(\frac{75}{x^2}) \\
&= 2x^2 + \frac{4x \cdot 75}{x^2} \\
&= 2x^2 + \frac{300x}{x^2} \\
&= 2x^2 + 300 \cdot \frac{x}{x^2}
\end{align}$$
However, this has been marked incorrect. What's the issue with my process?
Best Answer
It is the volume of the object that is $LWH$ and is given in cm$^3$. If you mean the volume is $75$ cm$^3$ the surface area is $2x^2+4xH$, which is $2x^2+\frac {300}x$ as you say. Maybe they expected you to replace $\frac x{x^2}$ with $\frac 1x$