Question 1169787
Let's break down this navigation problem step-by-step.

**Understanding the Problem**

We have two navigation stations A and B, and an airplane flying on a parallel course. We're given the distance between A and B, the airplane's distance from the line AB, the time difference in signal reception, and the signal speed. We need to find the airplane's position.

**Diagram**

1.  Draw a horizontal line representing the line passing through stations A and B.
2.  Mark point B on the left and point A on the right, with a distance of 88 miles between them.
3.  Draw a horizontal line 66 miles above the line AB, representing the airplane's path.
4.  Let P be the position of the airplane.
5.  Draw lines PA and PB.
6.  Draw a line from P perpendicular to AB, intersecting AB at point C.

**Given Information**

* Distance AB = 88 miles
* Airplane's distance from line AB (PC) = 66 miles
* Time difference (Δt) = 350 microseconds
* Signal speed (v) = 0.2 mile/microsecond

**Solution**

1.  **Distance Difference:**
    * The difference in distances traveled by the signals is:
        * Δd = v * Δt = 0.2 miles/microsecond * 350 microseconds = 70 miles
    * Therefore, PB - PA = 70 miles.

2.  **Coordinates:**
    * Let B be the origin (0, 0).
    * Then A is at (88, 0).
    * Let P be at (x, 66).
    * Then C is at (x, 0).

3.  **Distances PA and PB:**
    * PA = √((x - 88)² + 66²)
    * PB = √(x² + 66²)

4.  **Equation:**
    * PB - PA = 70
    * √(x² + 66²) - √((x - 88)² + 66²) = 70

5.  **Solve for x:**
    * √(x² + 66²) = 70 + √((x - 88)² + 66²)
    * Square both sides:
        * x² + 66² = 4900 + 140√((x - 88)² + 66²) + (x - 88)² + 66²
        * x² = 4900 + 140√((x - 88)² + 66²) + x² - 176x + 88²
        * 176x - 4900 - 88² = 140√((x - 88)² + 66²)
        * 176x - 4900 - 7744 = 140√((x - 88)² + 4356)
        * 176x - 12644 = 140√((x - 88)² + 4356)
        * (176x - 12644) / 140 = √((x - 88)² + 4356)
        * (44x - 3161) / 35 = √((x - 88)² + 4356)
    * Square both sides again:
        * (44x - 3161)² / 35² = (x - 88)² + 4356
        * (1936x² - 278128x + 9992081) / 1225 = x² - 176x + 7744 + 4356
        * (1936x² - 278128x + 9992081) / 1225 = x² - 176x + 12100
        * 1936x² - 278128x + 9992081 = 1225(x² - 176x + 12100)
        * 1936x² - 278128x + 9992081 = 1225x² - 215600x + 14822500
        * 711x² - 62528x - 4830419 = 0
    * Solve the quadratic equation using the quadratic formula:
        * x = (62528 ± √(62528² - 4 * 711 * (-4830419))) / (2 * 711)
        * x = (62528 ± √(3909775384 + 13739773956)) / 1422
        * x = (62528 ± √17649549340) / 1422
        * x = (62528 ± 132851.61) / 1422
        * x ≈ 136.69 or x ≈ -48.68

    * Since the plane is traveling east, we choose the positive solution.
    * x ≈ 136.69

6.  **Position of the Plane:**
    * The plane's position is approximately (136.69, 66).
    * This means the plane is approximately 136.69 miles east of station B.

**Final Answer**

The plane is approximately 136.69 miles east of station B.