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A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is 25 degrees. From a point 1792 feet closer to the mountain along the plain, they find that the angle of elevation is 28 degrees. How high (in feet) is the mountain?

A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is 25 degrees. From a point 1792 feet closer to the mountain along the plain, they find that the angle of elevation is 28 degrees. 
How high (in feet) is the mountain?

Grade:12

4 Answers

Vikas TU
14149 Points
9 years ago
I don't think that converting to radians will really help you. 

If you wanted to convert to radians, know that there are π radians in 180 degrees, so make a ratio: 

π/180 = x/25 

solve for x: 

180x = 25π 
x = 5π/36 

But in this case, to solve for the height of the mountain, we know we can draw a line from the top of the mountain to the ground to make a right triangle. So we're dealing with the base and the height of a right triangle so we can use tangent. 

Charting out the diagram below, we have an unknown height and an unknown base, but we know the partial distance of the base (the value between the two angles). 
....................
Vikas TU
14149 Points
9 years ago
continue.....................
 
so we can set up two tangent functions: 

tan(x) = opp/adj 
tan(25) = x / (1792 + y) 
tan(28) = x / y 

Now we have a system of equations. Let's solve for the second equation to be y in terms of x, then substitute and solve for x: 

tan(28) = x / y 
y = x / tan(28) 

tan(25) = x / (1792 + y) 
tan(25) = x / (1792 + x / tan(28)) 

in the denominator of the right side, make a common denominator so we can do the addition: 

tan(25) = x / (1792 tan(28) / tan(28) + x / tan(28)) 
Vikas TU
14149 Points
9 years ago
continue............
Now add the numerators: 

tan(25) = x / [(1792 tan(28) + x) / tan(28)] 

Now turn the division of fractions on the right side to a multiplication of fractions using reciprocal: 

tan(25) = x * [tan(28) / (1792 tan(28) + x)] 
tan(25) = x tan(28) / (1792 tan(28) + x) 

multiply both sides by the denominator: 

tan(25) (1792 tan(28) + x) = x tan(28) 

Multiply tan(25) through both terms inside the parenthesis: 

1792 tan(25) tan(28) + x tan(25) = x tan(28) 

Now subtract both sides by x tan(25): 

1792 tan(25) tan(28) = x tan(28) - x tan(25) 

Factor out the x, then divide both sides by the result: 
 
Vikas TU
14149 Points
9 years ago
continue.....................
 
1792 tan(25) tan(28) = x [tan(28) - tan(25)] 

x = 1792 tan(25) tan(28) / [tan(28) - tan(25)] 

Now using a calculator, we can solve for x: 

x = 6793.53 ft (rounded to 2 DP)

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