$$ \frac{(36.54)^2 - (3.46)^2}{x} = 40 $$

### Concept & Formula The equation asks us to solve for an unknown denominator by first simplifying the numerator using the difference of squares identity. $$a^2 - b^2 = (a - b)(a + b)$$ ### Step-by-Step Solution * Let the unknown value be $x$. * Focus on the numerator: $(36.54)^2 - (3.46)^2$. * Apply the formula where $a = 36.54$ and $b = 3.46$: $$(36.54 - 3.46)(36.54 + 3.46)$$ * Calculate the sum: $36.54 + 3.46 = 40.00$. * Calculate the difference: $36.54 - 3.46 = 33.08$. * Substitute these back into the equation: $$ \frac{33.08 \times 40}{x} = 40 $$ * The $40$ on the left side of the numerator cancels out with the $40$ on the right side of the equation. * This leaves $ \frac{33.08}{x} = 1 $, which means $x = 33.08$. ### Exam Strategy & Shortcut Realize that the sum of the two numbers $(36.54 + 3.46)$ is exactly $40$. Because the right-hand side of the equation is also $40$, it means the $(a+b)$ component is canceled out. Therefore, $x$ must be exactly equal to the $(a-b)$ component. Just calculate $36.54 - 3.46 = 33.08$. ### Common Pitfall Misplacing the decimal point during the subtraction step. Looking at the options, $3.308$, $33.08$, and $330.8$ are provided specifically to trap students who misalign their decimals. ### Final Answer Therefore, the correct answer is 33.08.