SAT Question f243c383: Area and Volume answer and explanation with Desmos solution

Choice B is correct. Let $x$ represent the side length, in $\text{cm}$, of each square base. If the two prisms are glued together along a square base, the resulting prism has a surface area equal to twice the surface area of one of the prisms, minus the area of the two square bases that are being glued together, which yields $2K-2x^2{\text{cm}}^2$. It's given that this resulting surface area is equal to $\frac{92}{47}K{\text{cm}}^2$, so $2K-2x^2=\frac{92}{47}K$. Subtracting $\frac{92}{47}K$ from both sides of this equation yields $2K-\frac{92}{47}K-2x^2=0$. This equation can be rewritten by multiplying $2K$ on the left-hand side by $\frac{47}{47}$, which yields $\frac{94}{47}K-\frac{92}{47}K-2x^2=0$, or $\frac{2}{47}K-2x^2=0$. Adding $2x^2$ to both sides of this equation yields $\frac{2}{47}K=2x^2$. Multiplying both sides of this equation by $\frac{47}{2}$ yields $K=47x^2$. The surface area $K$, in ${\text{cm}}^2$, of each rectangular prism is equivalent to the sum of the areas of the two square bases and the areas of the four lateral faces. Since the height of each rectangular prism is $90$ cm and the side length of each square base is $x$ cm, it follows that the area of each square base is $x^2{\text{cm}}^2$ and the area of each lateral face is $90x{\text{cm}}^2$. Therefore, the surface area of each rectangular prism can be represented by the expression $2x^2+4\left(90x\right)$, or $2x^2+360x$. Substituting this expression for $K$ in the equation $K=47x^2$ yields $2x^2+360x=47x^2$. Subtracting $2x^2$ and $360x$ from both sides of this equation yields $0=45x^2-360x$. Factoring $x$ from the right-hand side of this equation yields $0=x\left(45x-360\right)$. Applying the zero product property, it follows that $x=0$ and $45x-360=0$. Adding $360$ to both sides of the equation $45x-360=0$ yields $45x=360$. Dividing both sides of this equation by $45$ yields $x=8$. Since a side length of a rectangular prism can't be $0$, the length of each square base is $8$ cm.