SOLUTION: From a point A at the foot of the mountain, the angle of elevation of the top B is 60°. After ascending the mountain one mile at an inclination of 30° to the horizon and re

Question 1182691: From a point A at the foot of the mountain, the angle
of elevation of the top B is 60°. After ascending the
mountain one mile at an inclination of 30° to the horizon
and reaching a point C, observer finds that the angle ACB
is 135°. Compute the height of the mountain.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
my diagram of what i think is your problem and its solution zis shown below:

the first triangle formed is triangle ABD.
angle BAD is equal to 60 degrees.

angle BCA = 135 degrees.
angle BAC = 30 degrees
angle CAD = 30 degrees.

the length of AC is equal to 1 mile.

the new triangle formed is triangle BCA.
since the sum of the angles of a triangle is 180 degrees, angle ABC is equal to 180 minus 135 minus 30 = 15 degrees.

using the law of sines, we can solve the length of AB as follows:

1/sin(15) = AB/sin(135)
solve for AB to get:
AB = sin(135)/sin(15) = 2.732050808.

sin(60) = BD / BA.
solve for BD to get:
BD = BA * sin(60) = 2.732050808 * sin(60) = 2.366025404.
that's the height of the mountain.

to confirm that i did this correctly, i solved for AD.

cos(60) = AD / BA.
solve for AD to get:
AD = BA * cos(60) = 2.732050808 * cos(60) = 1.366025404.

by the pythagorus formula, AB^2 = BD^2 + AD^2.
that becomes:
2.732050808^2 = 2.366025404^2 + 1.366025404^2 which becomes:
2.732050808^2 = 2.732050808^2, confirming that we have the right measurements for triangle ABD.

this could only happen if we had the right measurements for AB.

i believe this is correct, based on my understanding of the problem.

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