Matt measures the angle of elevation of the peak of a mountain as 35 degrees. Susie, who is 1200 feet closer on a straight level path, measures the angle of elevation as 42 degrees. What is the distance from Matt to the peak of the mountain?
(Use law of Sines, AAS)
Let x be the distance from Matt to the peak of the mountain.
Using the Law of Sines, we have:
sin(35°) / x = sin(42°) / (x - 1200)
Solving for x, we get:
x = (1200 * sin(35°)) / (sin(42°) - sin(35°))
x ≈ 1645.7 feet
To find the distance from Matt to the peak of the mountain, we can use the law of sines and the given information.
Let's label the distance from Matt to the peak of the mountain as x.
Using the law of sines, we have:
sin(35°) / x = sin(42°) / (x + 1200)
Now we can solve for x:
Cross multiplying, we get:
sin(35°) * (x + 1200) = sin(42°) * x
Distributing, we have:
x * sin(35°) + 1200 * sin(35°) = x * sin(42°)
Rearranging the equation, we get:
x * (sin(42°) - sin(35°)) = 1200 * sin(35°)
Dividing both sides by (sin(42°) - sin(35°)), we get:
x = (1200 * sin(35°)) / (sin(42°) - sin(35°))
Now we can calculate this value:
x = (1200 * 0.5736) / (0.6691 - 0.5736)
x = 688.32 / 0.0955
x ≈ 7195.79
Therefore, the distance from Matt to the peak of the mountain is approximately 7195.79 feet.
To solve this problem using the law of sines and AAS (Angle-Angle-Side) postulate, we can set up a triangle using the given information.
Let's assume that Matt is at point M, Susie is at point S, and the peak of the mountain is at point P. We have two angles: angle M and angle S, which correspond to the angles of elevation measured by Matt and Susie, respectively.
From the given information, we know that angle M (measured by Matt) is 35 degrees, and angle S (measured by Susie) is 42 degrees. We can also determine that angle P is 180 degrees minus the sum of angles M and S, since the sum of angles in any triangle is 180 degrees. Therefore, angle P = 180 - (35 + 42) = 103 degrees.
We can now set up the proportion using the law of sines:
(MP / sin S) = (MS / sin P)
Where MP is the distance from Matt to the peak, MS is the distance from Matt to Susie (1200 feet), and sin S is sin 42 degrees.
Substituting the known values into the equation, we have:
(MP / sin 42) = (1200 / sin 103)
To find MP, we can multiply both sides of the equation by sin 42:
MP = (1200 / sin 103) * sin 42
Using a calculator, evaluate the right side of the equation:
MP ≈ (1200 / 0.6970) * 0.6691
MP ≈ 1200 * 0.959 ≈ 1,151.2 feet
Therefore, the distance from Matt to the peak of the mountain is approximately 1,151.2 feet.