In the xy-plane, the coordinates of point P are \((1.2, -0.5)\). The coordinates of point Q are \((4.7, \mathrm{y})\), where...

1. TRANSLATE the problem information

• Given information:
  • Point P has coordinates \((1.2, -0.5)\)
  • Point Q has coordinates \((4.7, y)\) where y is positive
  • Distance between P and Q is 3.7 units
• We need to find the value of y

2. INFER the approach

• Since we have two points and the distance between them, we should use the distance formula
• The distance formula will create an equation with y as the unknown

3. TRANSLATE the distance formula

• Distance formula: \(\mathrm{d} = \sqrt{(\mathrm{x}_2 - \mathrm{x}_1)^2 + (\mathrm{y}_2 - \mathrm{y}_1)^2}\)
• Substituting our values: \(3.7 = \sqrt{(4.7 - 1.2)^2 + (y - (-0.5))^2}\)
• Simplifying: \(3.7 = \sqrt{(3.5)^2 + (y + 0.5)^2}\)

4. SIMPLIFY the equation

• Calculate: \(3.7 = \sqrt{12.25 + (y + 0.5)^2}\)
• Square both sides to eliminate the square root: \(3.7^2 = 12.25 + (y + 0.5)^2\)
• Calculate: \(13.69 = 12.25 + (y + 0.5)^2\) (use calculator)
• Isolate the squared term: \((y + 0.5)^2 = 13.69 - 12.25 = 1.44\)

5. SIMPLIFY further to solve for y

• Take the square root of both sides: \(y + 0.5 = ±\sqrt{1.44}\)
• Calculate: \(y + 0.5 = ±1.2\) (use calculator)
• This gives us two possibilities:
  • \(y + 0.5 = 1.2 → y = 0.7\)
  • \(y + 0.5 = -1.2 → y = -1.7\)

6. APPLY CONSTRAINTS to select the final answer

• The problem states that y is positive
• Since -1.7 is negative, we reject this solution
• Therefore: \(y = 0.7\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when working with the distance formula, especially when squaring both sides or dealing with the nested operations.

Common mistakes include calculating \(3.7^2\) incorrectly, making sign errors with \((y + 0.5)\), or errors in isolating the squared term. These computational errors lead to incorrect values and cause students to select wrong answer choices or become confused about which approach to take.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students find both \(y = 0.7\) and \(y = -1.7\) but fail to recognize that the problem specifies y must be positive.

Without applying this constraint, they might select the wrong answer or become uncertain about which solution is correct. This may lead them to select Choice B (1.2) if they confuse the constraint with the calculated square root value, or guess randomly between the positive solutions.

The Bottom Line:

This problem requires careful algebraic manipulation of the distance formula combined with attention to the given constraints. Success depends on systematic execution of each step rather than rushing through the calculations.