Frances stands $15$ meters directly south of a locked gate in a fence that runs east-west. Immediately behind the fence is a box of chocolates, located $x$ meters east of the locked gate. An unlocked gate lies $9$ meters east of the box, and another unlocked gate lies $8$ meters west of the locked gate. Frances can reach the box by walking toward an unlocked gate, passing through it, and walking toward the box. It happens that the total distance Frances would travel is the same via either unlocked gate. What is the value of $x$?
$$\textbf{(A) } 3\frac{2}{7} \qquad \textbf{(B) } 3\frac{3}{7} \qquad \textbf{(C) } 3\frac{4}{7} \qquad \textbf{(D) } 3\frac{5}{7} \qquad \textbf{(E) } 3\frac{6}{7}$$
Difficulty: 34/100 - A computation heavy problem whose wording causes confusion.
Core Concepts: Pythagorean theorem, right triangles, radical simplification
Challenges: Comprehending the problem may be hard to get past, and equating the two path lengths and solving via squaring may be difficult for less experienced contestants.
Techniques: Assigning coordinates or employing the Pythagorean theorem, and solving the resulting linear equation.
Error-prone Steps: Misinterpreting wording and creating incorrect diagrams, miscalculating each length, or computation errors in quadratic solving.
Ideal Time:
Experienced: ≤ 2-3 min
Intermediate: ≤ 3-5 min
Beginner: ≤ 7 min