SOLUTION: Hello! I have this word problem, and we haven't learned the Sine Theorem yet in class, so we can't use it. I have tried EVERYTHING else, and I can't find any other way. Bob is d

Algebra ->  Trigonometry-basics -> SOLUTION: Hello! I have this word problem, and we haven't learned the Sine Theorem yet in class, so we can't use it. I have tried EVERYTHING else, and I can't find any other way. Bob is d      Log On


   

Question 692115: Hello! I have this word problem, and we haven't learned the Sine Theorem yet in class, so we can't use it. I have tried EVERYTHING else, and I can't find any other way.
Bob is driving along a straight and level road straight toward a mountain. At some point on his trip he measures the angle of elevation to the top of the mountain and finds it to be 23 degrees and 10 minutes. He then drives 1 mile (i mile=5280 ft) more and measures the angle of elevation to be 32 degrees and 17 minutes. Find the height of the mountain to the nearest foot.
If you could help, that would be fantastic.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
A diagram may help make sense of this:

Angle BCD is 90 degrees. This makes triangles BAC and BDC right triangles.
Angle BAD is 23 degrees 10 minutes (or 23.1666 degrees).
Angle BDC is 32 degrees 17 minutes (or 32.2833 degrees).
AD is 1 mile.
BC is the height of the mountain. Let's call it "x".
We will need to refer to DC also. Let's call it "y".

In triangle BAC the vertical side, x, is opposite to angle BAC and the horizontal side, 1+y, is adjacent. Since tan is the ratio of opposite over adjacent:
tan%2823.1666%29+=+x%2F%281%2By%29

In triangle BDC the vertical side, x, is opposite to angle BAC and the horizontal side, y, is adjacent. Since tan is the ratio of opposite over adjacent:
tan%2832.2833%29+=+x%2Fy

Solving the last equation for y...
y%2Atan%2832.2833%29+=+x
y+=+x%2Ftan%2832.2833%29

Substituting this expression for y into the first equation:
tan%2823.1666%29+=+x%2F%281%2Bx%2Ftan%2832.2833%29%29
Now we solve for x. Multiplying the numerator and denominator of the fraction on the right side by tan(32.2833):

which simplifies to:
tan%2823.1666%29+=+%28x%2Atan%2832.2833%29%29%2F%28tan%2832.2833%29%2Bx%29
Multiplying both sides by the denominator on the right side:

which simplifies to:
tan%2832.2833%29%2Atan%2823.1666%29%2Bx%2Atan%2823.1666%29+=+x%2Atan%2832.2833%29
Next we gather the terms with x on the right side by subtracting x*tan(23.1666) from both sides:
tan%2832.2833%29%2Atan%2823.1666%29+=+x%2Atan%2832.2833%29+-+x%2Atan%2823.1666%29
Factoring out x on the right side:
tan%2832.2833%29%2Atan%2823.1666%29+=+x%2A%28tan%2832.2833%29+-+tan%2823.1666%29%29
Then we divide both sides by (tan(32.2833) - tan(23.1666)):
%28tan%2832.2833%29%2Atan%2823.1666%29%29%2F%28tan%2832.2833%29+-+tan%2823.1666%29%29+=+x
Last of all we use our calculators to find the two tan's:
%280.6453%2A0.4279%29%2F%280.6453+-+0.4279%29+=+x
0.2742%2F0.2180+=+x
1.2578+=+x
This is the height of the mountain, in miles. To get the height in feet, just multiply this by 5280.

Note: The angles, the tan's and some of the calculations were all rounded to four decimal places. So there is a small error in the answer. If you round to more than just 4 decimal places then you will get a more accurate answer.