[Geometry]
Problem: One of the faces of a rectangular prism has area $27 \mathrm{~cm}^{2}$. Another face has area $32 \mathrm{~cm}^{2}$. If the volume of the prism is $144 \mathrm{~cm}^{3}$, determine the surface area of the prism in $\mathrm{cm}^{2}$.

Solution: Suppose that the rectangular prism has dimensions $a \mathrm{~cm}$ by $b \mathrm{~cm}$ by $c \mathrm{~cm}$.

Suppose further that one of the faces that is $a \mathrm{~cm}$ by $b \mathrm{~cm}$ is the face with area $27 \mathrm{~cm}^{2}$ and that one of the faces that is $a \mathrm{~cm}$ by $c \mathrm{~cm}$ is the face with area $32 \mathrm{~cm}^{2}$. (Since every pair of non-congruent faces shares exactly one side length, there is no loss of generality in picking these particular variables for these faces.)

Therefore, $a b=27$ and $a c=32$.

Further, we are told that the volume of the prism is $144 \mathrm{~cm}^{3}$, and so $a b c=144$.



Thus, $b c=\frac{a^{2} b^{2} c^{2}}{a^{2} b c}=\frac{(a b c)^{2}}{(a b)(a c)}=\frac{144^{2}}{(27)(32)}=24$.

(We could also note that $a b c=144$ means $a^{2} b^{2} c^{2}=144^{2}$ or $(a b)(a c)(b c)=144^{2}$ and so $b c=\frac{144^{2}}{(27)(32)}$.)

In other words, the third type of face of the prism has area $24 \mathrm{~cm}^{2}$.

Thus, since the prism has two faces of each type, the surface area of the prism is equal to $2\left(27 \mathrm{~cm}^{2}+32 \mathrm{~cm}^{2}+24 \mathrm{~cm}^{2}\right)$ or $166 \mathrm{~cm}^{2}$.
Suppose that the rectangular prism has dimensions $a \mathrm{~cm}$ by $b \mathrm{~cm}$ by $c \mathrm{~cm}$.

Suppose further that one of the faces that is $a \mathrm{~cm}$ by $b \mathrm{~cm}$ is the face with area $27 \mathrm{~cm}^{2}$ and that one of the faces that is $a \mathrm{~cm}$ by $c \mathrm{~cm}$ is the face with area $32 \mathrm{~cm}^{2}$. (Since every pair of non-congruent faces shares exactly one side length, there is no loss of generality in picking these particular variables for these faces.)

Therefore, $a b=27$ and $a c=32$.

Further, we are told that the volume of the prism is $144 \mathrm{~cm}^{3}$, and so $a b c=144$.

Since $a b c=144$ and $a b=27$, then $c=\frac{144}{27}=\frac{16}{3}$.

Since $a b c=144$ and $a c=32$, then $b=\frac{144}{32}=\frac{9}{2}$.

This means that $b c=\frac{16}{3} \cdot \frac{9}{2}=24$.

In $\mathrm{cm}^{2}$, the surface area of the prism equals $2 a b+2 a c+2 b c=2(27)+2(32)+2(24)=166$. Thus, the surface area of the prism is $166 \mathrm{~cm}^{2}$.

Final answer: $166$ $cm^2$