a surveyor finds that from Point A on the ground the angle of elevation to the top of a mountin is 23 degrees. when he is at point B that is 1/4 closer to the base of the mountain the angle of elevation is 43 degrees. how high is the moutnain? One mile =5280 feet. (Assume that thr base of the mountain and the two observation points are on the same line.)Give answer to the nearest foot.
tan23 = Y/X,
Y = Xtan23,
tan43 = Y/(X-.25).
Y = (X-0.25)tan43.
y = Xtan23 = (X-0.25)tan43,
Xtan23 = (X-0.25)tan43,
Xtan23 = Xtan43-0.25tan43,
0.4245X = 0.9325X-0.2331,
0.4245X-0.9325X = -0.2331,
-0.508X = -0.2331,
X = 0.4589 Miles.
y = 0.4589tan23 = 0.1948 miles.
Y = 0.1948mi * 5280ft/mi = 1029 Ft.
= Height of mountain.
To find the height of the mountain, we can use trigonometry. Let's denote the height of the mountain as "h" and the distance from Point A to the base of the mountain as "x".
From Point A, the angle of elevation to the top of the mountain is given as 23 degrees. This means that the opposite side (height of the mountain "h") is the unknown, and the adjacent side (distance "x") is known.
We can use the tangent function to find the height of the mountain at Point A:
tan(23 degrees) = h / x
Next, let's consider Point B which is 1/4 closer to the base of the mountain. This means that the distance from Point B to the base of the mountain is 3/4 of the distance from Point A to the base of the mountain. So, the distance from Point B to the base of the mountain is (3/4)x.
From Point B, the angle of elevation to the top of the mountain is given as 43 degrees. Again, the opposite side (height of the mountain "h") is the unknown, and the adjacent side (distance "3/4x") is known.
We can use the tangent function again to find the height of the mountain at Point B:
tan(43 degrees) = h / (3/4x)
Now we have two equations:
1) tan(23 degrees) = h / x
2) tan(43 degrees) = h / (3/4x)
We can solve this system of equations to find the value of "h". Rearranging equation 1, we get:
h = x * tan(23 degrees)
Substituting this value of "h" into equation 2, we have:
tan(43 degrees) = (x * tan(23 degrees)) / (3/4x)
Now we can solve for "x". Dividing both sides of the equation by (3/4), we get:
tan(43 degrees) / (3/4) = x * tan(23 degrees)
x = tan(43 degrees) / (3/4 * tan(23 degrees))
Calculating this value, we find:
x ≈ 0.868
Now that we have the value of "x", we can substitute it back into equation 1 to find the height of the mountain at Point A:
h = x * tan(23 degrees)
= 0.868 * tan(23 degrees)
Calculating this value, we find:
h ≈ 0.391
Finally, we convert this height to feet by multiplying by 5280 since 1 mile = 5280 feet:
Height of the mountain ≈ 0.391 * 5280 ≈ 2060 feet
Therefore, the height of the mountain is approximately 2060 feet.