A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is
26
∘
. From a point 2000 feet closer to the mountain along the plain, they find that the angle of elevation is
28
∘
.
How high (in feet) is the mountain?
To find the height of the mountain, we can use trigonometry and set up two right triangles.
Let's assume that the height of the mountain is represented by "h" in feet.
From the first observation point, we have a right triangle formed by the observer, the top of the mountain, and the base of the mountain. The angle of elevation is 26 degrees.
Using trigonometry, we can set up the following equation:
tan(26 degrees) = h / x
Where x represents the distance between the observer and the mountain.
From the second observation point, which is 2000 feet closer to the mountain, we have another right triangle formed. The angle of elevation is 28 degrees.
Using trigonometry again, we can set up another equation:
tan(28 degrees) = h / (x - 2000)
Now we have a system of two equations with two unknowns. Let's solve it:
First, let's rewrite the equations using the tangent function:
h / x = tan(26 degrees) ---- (1)
h / (x - 2000) = tan(28 degrees) ---- (2)
Now we can solve for h by isolating it in one of the equations. Let's isolate h in equation (1):
h = x * tan(26 degrees)
We can substitute this expression for h in equation (2):
(x * tan(26 degrees)) / (x - 2000) = tan(28 degrees)
Now we can solve for x:
(x * tan(26 degrees)) = tan(28 degrees) * (x - 2000)
x * tan(26 degrees) = tan(28 degrees) * x - 2000 * tan(28 degrees)
x * tan(26 degrees) - tan(28 degrees) * x = -2000 * tan(28 degrees)
x * (tan(26 degrees) - tan(28 degrees)) = -2000 * tan(28 degrees)
x = (-2000 * tan(28 degrees)) / (tan(26 degrees) - tan(28 degrees))
Now we can substitute the value of x back into equation (1) to find the height, h:
h = x * tan(26 degrees)
Let's calculate the values:
tan(26 degrees) ≈ 0.4877
tan(28 degrees) ≈ 0.5317
Substituting these values:
x = (-2000 * 0.5317) / (0.4877 - 0.5317)
x ≈ 8358.26 feet
Now we can substitute this value of x back into equation (1):
h = 8358.26 * tan(26 degrees)
h ≈ 8358.26 * 0.4877
h ≈ 4072.92 feet
Therefore, the height of the mountain is approximately 4072.92 feet.
To find the height of the mountain, we can use trigonometry and the given information about the angles of elevation.
Let's assign some variables to the problem:
- Let h be the height of the mountain.
- Let d be the distance from the initial point on the plain to the mountain.
- Let d' be the distance from the second point (2000 feet closer) on the plain to the mountain.
Now, we can use the tangent function to set up equations based on the angles of elevation.
Using the first observation:
tan(26°) = h/d
And using the second observation:
tan(28°) = h/(d - 2000)
We have two equations with two unknowns (h and d). We can solve this system of equations to find the values.
First, let's solve the first equation for d:
d = h/tan(26°)
Substitute this into the second equation:
tan(28°) = h/(h/tan(26°) - 2000)
Now, we can solve for h:
tan(28°) = h/((h/tan(26°)) - 2000)
To solve this equation, we can multiply both sides by the denominator:
tan(28°) * (h/tan(26°)) - 2000 * tan(28°) = h
Rearranging the equation to isolate h:
h * (tan(28°)/tan(26°) - 1) = 2000 * tan(28°)
h = (2000 * tan(28°)) / (tan(28°)/tan(26°) - 1)
Using a calculator, we can evaluate this expression to find the value of h.
Plug in the values for the tangent of angles 26° and 28°, and calculate:
h ≈ (2000 * 0.531) / (0.531/0.486 - 1) ≈ 1347.92 feet
Therefore, the height of the mountain is approximately 1347.92 feet.
did you make a sketch??
On mine I labelled the top of the mountain as P and its perpendicular base pont as Q.
Farthest point as A and the closer point as B, AB = 2000
Look at triangle ABP
angle A=26, angle ABP = 152, then angle APB = 2
by the sine law,
BP/sin26 = 2000/sin2
BP = 2000sin26/sin2 = ....
Then in the right-angled triangle BPQ
sin28 = PQ/BP
PQ = BPsin28 = ....
Tan26 = h/AC.
h = AC*Tan26.
Tan28 = h/(AC-2000),
h = (AC-2000)Tan28.
AC*Tan26 = (AC-2000)Tan28,
AC = (AC-2000)1.09,
0.92AC = AC-2000,
0.08AC = 2000.
AC = 25,000 ft.
Tan26 = h/25,000,
h = 12,200 Ft.