From a point on ground level, you measure the angle of elevation to the top of a moutain to be 38 degrees. Then you walk 200 m fartheraway from the mountain and find that the angle of elevation is now 20 degrees. Find the height of the mountain. Round the answer to the nearest meter.
Let h be the height of the mountain, x be the distance from the base to the first observation point.
tan 38= h/x
or x= h ctn 38
Second point:
Tan 20= h/(200+x)
h= tan38 * (200 + h ctn30)
h-h tan38 ctn20 = tan38 *200
solve for h.
To solve the problem, we will use trigonometry and set up two equations.
Let h be the height of the mountain and x be the distance from the base to the first observation point.
First observation point:
Using the angle of elevation of 38 degrees, we can set up the following equation:
tan(38) = h / x
Second observation point:
Since you walked 200 meters further away from the mountain, the distance from the base to the second observation point is x + 200.
Using the angle of elevation of 20 degrees, we can set up the following equation:
tan(20) = h / (x + 200)
Now, we can solve the equations to find the height of the mountain.
Rearranging the first equation, we get:
x = h / tan(38)
Substituting this value of x into the second equation, we get:
tan(20) = h / (h / tan(38) + 200)
Simplifying this equation further, we get:
tan(20) = tan(38) * (h / (h + 200tan(38)))
Now, we can solve this equation for h.
h - h tan(38) tan(20) = tan(38) * 200
Solving for h, we get:
h = (tan(38) * 200) / (1 - tan(38) tan(20))
Calculating the value using a calculator, we find that h is approximately 295 meters.
Therefore, the height of the mountain is approximately 295 meters (rounded to the nearest meter).