document.write( "Question 1169787: Two long-range navigation stations A and B lie on a line running east and west,
\n" ); document.write( "and A is 88 miles due east of B. An airplane is travelling east on a straight line course
\n" ); document.write( "that is 66 miles north of the line tough A and B. Signals are sent at the same time
\n" ); document.write( "from A and B, and the signal from A reaches the plane 350 microseconds before
\n" ); document.write( "the one from B. If the signals travel at the rate of 0.2 mile/microsecond, locate the
\n" ); document.write( "position of the plane. \n" ); document.write( "
Algebra.Com's Answer #851303 by CPhill(1959)\"\" \"About 
You can put this solution on YOUR website!
Let's break down this navigation problem step-by-step.\r
\n" ); document.write( "\n" ); document.write( "**Understanding the Problem**\r
\n" ); document.write( "\n" ); document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "**Diagram**\r
\n" ); document.write( "\n" ); document.write( "1. Draw a horizontal line representing the line passing through stations A and B.
\n" ); document.write( "2. Mark point B on the left and point A on the right, with a distance of 88 miles between them.
\n" ); document.write( "3. Draw a horizontal line 66 miles above the line AB, representing the airplane's path.
\n" ); document.write( "4. Let P be the position of the airplane.
\n" ); document.write( "5. Draw lines PA and PB.
\n" ); document.write( "6. Draw a line from P perpendicular to AB, intersecting AB at point C.\r
\n" ); document.write( "\n" ); document.write( "**Given Information**\r
\n" ); document.write( "\n" ); document.write( "* Distance AB = 88 miles
\n" ); document.write( "* Airplane's distance from line AB (PC) = 66 miles
\n" ); document.write( "* Time difference (Δt) = 350 microseconds
\n" ); document.write( "* Signal speed (v) = 0.2 mile/microsecond\r
\n" ); document.write( "\n" ); document.write( "**Solution**\r
\n" ); document.write( "\n" ); document.write( "1. **Distance Difference:**
\n" ); document.write( " * The difference in distances traveled by the signals is:
\n" ); document.write( " * Δd = v * Δt = 0.2 miles/microsecond * 350 microseconds = 70 miles
\n" ); document.write( " * Therefore, PB - PA = 70 miles.\r
\n" ); document.write( "\n" ); document.write( "2. **Coordinates:**
\n" ); document.write( " * Let B be the origin (0, 0).
\n" ); document.write( " * Then A is at (88, 0).
\n" ); document.write( " * Let P be at (x, 66).
\n" ); document.write( " * Then C is at (x, 0).\r
\n" ); document.write( "\n" ); document.write( "3. **Distances PA and PB:**
\n" ); document.write( " * PA = √((x - 88)² + 66²)
\n" ); document.write( " * PB = √(x² + 66²)\r
\n" ); document.write( "\n" ); document.write( "4. **Equation:**
\n" ); document.write( " * PB - PA = 70
\n" ); document.write( " * √(x² + 66²) - √((x - 88)² + 66²) = 70\r
\n" ); document.write( "\n" ); document.write( "5. **Solve for x:**
\n" ); document.write( " * √(x² + 66²) = 70 + √((x - 88)² + 66²)
\n" ); document.write( " * Square both sides:
\n" ); document.write( " * x² + 66² = 4900 + 140√((x - 88)² + 66²) + (x - 88)² + 66²
\n" ); document.write( " * x² = 4900 + 140√((x - 88)² + 66²) + x² - 176x + 88²
\n" ); document.write( " * 176x - 4900 - 88² = 140√((x - 88)² + 66²)
\n" ); document.write( " * 176x - 4900 - 7744 = 140√((x - 88)² + 4356)
\n" ); document.write( " * 176x - 12644 = 140√((x - 88)² + 4356)
\n" ); document.write( " * (176x - 12644) / 140 = √((x - 88)² + 4356)
\n" ); document.write( " * (44x - 3161) / 35 = √((x - 88)² + 4356)
\n" ); document.write( " * Square both sides again:
\n" ); document.write( " * (44x - 3161)² / 35² = (x - 88)² + 4356
\n" ); document.write( " * (1936x² - 278128x + 9992081) / 1225 = x² - 176x + 7744 + 4356
\n" ); document.write( " * (1936x² - 278128x + 9992081) / 1225 = x² - 176x + 12100
\n" ); document.write( " * 1936x² - 278128x + 9992081 = 1225(x² - 176x + 12100)
\n" ); document.write( " * 1936x² - 278128x + 9992081 = 1225x² - 215600x + 14822500
\n" ); document.write( " * 711x² - 62528x - 4830419 = 0
\n" ); document.write( " * Solve the quadratic equation using the quadratic formula:
\n" ); document.write( " * x = (62528 ± √(62528² - 4 * 711 * (-4830419))) / (2 * 711)
\n" ); document.write( " * x = (62528 ± √(3909775384 + 13739773956)) / 1422
\n" ); document.write( " * x = (62528 ± √17649549340) / 1422
\n" ); document.write( " * x = (62528 ± 132851.61) / 1422
\n" ); document.write( " * x ≈ 136.69 or x ≈ -48.68\r
\n" ); document.write( "\n" ); document.write( " * Since the plane is traveling east, we choose the positive solution.
\n" ); document.write( " * x ≈ 136.69\r
\n" ); document.write( "\n" ); document.write( "6. **Position of the Plane:**
\n" ); document.write( " * The plane's position is approximately (136.69, 66).
\n" ); document.write( " * This means the plane is approximately 136.69 miles east of station B.\r
\n" ); document.write( "\n" ); document.write( "**Final Answer**\r
\n" ); document.write( "\n" ); document.write( "The plane is approximately 136.69 miles east of station B.
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