Jacqueline is flying a plane and is descending at a 14°14° angle towards a runway. If she can see a mountain behind her at a 55°55° angle that is 400400 yards away from the runway, how much further does she have to fly until she lands? Round to the nearest tenth.
Question
Jacqueline is flying a plane and is descending at a 14°14° angle towards a runway. If she can see a mountain behind her at a 55°55° angle that is 400400 yards away from the runway, how much further does she have to fly until she lands? Round to the nearest tenth.
Solution
To solve this problem, we can use the properties of alternate angles in a triangle.
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First, we need to find the angle of the mountain from the horizontal. Since Jacqueline is descending at a 14° angle, the angle from the horizontal to her line of sight is 90° - 14° = 76°.
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The angle from the horizontal to the mountain is given as 55°. Therefore, the angle between Jacqueline's line of sight and the line to the mountain is 76° - 55° = 21°.
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Now we have a triangle with one side of 400 yards (the distance from the runway to the mountain) and the angles of 21° and 69° (since the sum of angles in a triangle is 180°, the third angle is 180° - 21° - 90° = 69°).
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We can use the law of sines to find the distance from Jacqueline to the runway (which we'll call d). The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. Therefore, we have:
d / sin(69°) = 400 / sin(21°)
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Solving for d gives us:
d = (400 * sin(69°)) / sin(21°)
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Calculating this gives us a distance of approximately 1057.2 yards. However, the question asks for the answer rounded to the nearest tenth, so the final answer is 1057.2 yards.
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