angle of elevation of the top of a mountain to a man is 45degree.if we walk along a slope of 30 degree of the mountain 100 meters then the angle of elevation of the top of mountain becomes 60 degree.find the height of the mountain.

The line of sight of 45 deg angle is

drawn as the hyp. of a rt triangle.
The ht. of the mountain is the ver. side of the triangle. the hor. side
side = 100 + X. The line of sight of the 60 deg. angle represents the hyp.
of a 2nd triangle and the ht. of the
mountain is the ver. side of both triangles.

Tan(45) = h /(100+X),
h = (100 + X) * Tan(45),
.
Tan(60) = h / X,
h = x * Tan(60).

h = (100 + X) * Tan(45) = X * Tan(60),
Solve for X:
70.7 + 0.707X = 1.73X,
- 1.73X + 0.707X = - 70.7
X = 69 METERS.
h = 69 * TAN(60) = 119.5 Meters =
ht. of mountain.

To find the height of the mountain, we can use trigonometry and set up a right triangle.

Let's say the height of the mountain is represented by 'h' meters.

In the first scenario, the angle of elevation from the man to the top of the mountain is 45 degrees. This draws a right triangle where the opposite side is the height of the mountain and the adjacent side is the horizontal distance along the slope. Let's label the adjacent side as 'x1'.

In the second scenario, the angle of elevation from the man to the top of the mountain becomes 60 degrees after walking 100 meters along the slope. This forms another right triangle where the adjacent side is now equal to 'x1 + 100' meters.

Using trigonometry, we can write the equations for both scenarios:

Scenario 1:
tan(45°) = h / x1

Scenario 2:
tan(60°) = h / (x1 + 100)

Now, we can solve these equations simultaneously to find the height of the mountain 'h'.

Divide the second equation by the first equation:
tan(60°) / tan(45°) = h / (x1 + 100) * x1 / h

Simplifying the left side:
sqrt(3) / 1 = (x1/x1 + 100)

Cross-multiply:
sqrt(3) * (x1 + 100) = x1

Solving for x1:
sqrt(3) * x1 + 100sqrt(3) = x1

Rearranging the equation:
sqrt(3) * x1 - x1 = -100sqrt(3)

Combining like terms:
(x1 * sqrt(3)) - x1 = -100sqrt(3)

Factoring x1 out:
x1 * (sqrt(3) - 1) = -100sqrt(3)

Solving for x1:
x1 = -100sqrt(3) / (sqrt(3) - 1)

Now that we know x1, we can use it in the equation from Scenario 1 to solve for the height 'h'.

Using the first equation:
tan(45°) = h / x1

Substituting the value of x1:
tan(45°) = h / (-100sqrt(3) / (sqrt(3) - 1))

Simplifying the right side:
tan(45°) = -h * (sqrt(3) - 1) / (100sqrt(3))

Since tan(45°) = 1, we can simplify it further:
1 = -h * (sqrt(3) - 1) / (100sqrt(3))

Multiply both sides by (100sqrt(3)):
100sqrt(3) = -h * (sqrt(3) - 1)

Rearranging the equation to solve for h:
h = -100sqrt(3) / (sqrt(3) - 1)

Therefore, the height of the mountain is approximately -100sqrt(3) / (sqrt(3) - 1) meters.