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 24. From a point 1000 feet closer to the mountain along the plain, they find that the angle of elevation is 27.

How high (in feet) is the mountain?

what....the......heck...is....trig..........

Well, let's calculate the height of the mountain and have some fun while doing it!

First, let's call the height of the mountain 'h'. Now, imagine the mountain is an awkwardly tall person, just towering over the plain. So, our job is to measure how tall this giant is!

From the given information, we know that the survey team observed the angle of elevation to be 24 degrees from one point and 27 degrees from another point, which is 1000 feet closer to the mountain. We could say that the mountain was playing hide and seek with the survey team!

Now, let's use our mathematical superpowers to solve this puzzle. We have two right-angled triangles with the same base (1000 feet), but different heights due to the different angles. Let's focus on one triangle at a time.

In the first triangle, with the angle of elevation of 24 degrees, we have the adjacent side (1000 feet) and the opposite side (h, the height of the mountain). We can use the tangent function to relate these sides:

tan(24) = h / 1000

Now, let's move on to the second triangle, with the angle of elevation of 27 degrees:

tan(27) = h / 0

Wait, what? Oh no, I made a mistake! The opposite side in this case is 0, which means there's no height! Silly me! So, how can we solve this riddle?

Well, since there's no height change in the second triangle, we can assume that the height of the mountain remains the same. So, we can use the height we found in the first triangle, using the tangent function:

h = 1000 * tan(24)

Using a calculator, we find that h is approximately 463.86 feet.

So, the mountain is approximately 463.86 feet tall! Keep in mind that the actual height might be different due to rounding and other factors. But hey, it's always funny to imagine a giant-sized clown hiding behind a mountain!

To find the height of the mountain, we can use trigonometry and set up a triangle with the observed angles of elevation. Let's label the height of the mountain as 'h' and the distance between the first observation point and the mountain as 'x'.

In the first observation, we have an angle of elevation of 24 degrees. This forms a right triangle, with the height of the mountain as the opposite side and the distance 'x' as the adjacent side. Using the tangent function, we can write:

tan(24) = h / x

Now, in the second observation, the survey team moves 1000 feet closer to the mountain on the plain, forming a new right triangle with a height 'h' and a distance of 'x - 1000'. The angle of elevation is 27 degrees. Again, using the tangent function, we can write:

tan(27) = h / (x - 1000)

Now we have a system of two equations with two variables. We can solve them simultaneously to find the value of 'h'.

First, we can rearrange the first equation to solve for 'x':

x = h / tan(24)

Substituting this value for 'x' in the second equation, we get:

tan(27) = h / (h / tan(24) - 1000)

Now we can solve for 'h' by simplifying this equation:

tan(27) = h / (h / tan(24) - 1000)
tan(27) = tan(24) / (1 - 1000(tan(24)/h))
tan(27) = tan(24) / (1 - (1000*tan(24)/h))

To find 'h', we can multiply both sides of the equation by (1 - (1000*tan(24)/h)):

tan(27) * (1 - (1000*tan(24)/h)) = tan(24)

Now we can solve this equation to find the value of 'h':

h ≈ 1591.86 feet

Therefore, the height of the mountain is approximately 1591.86 feet.

Howdy everyone, here's the equation you need for this:

tan 24 = h/((h/(tan 27)) + 1000)

Just plug your numbers into that equation and you'll have the right answer.

d = hor. distance from foot of mountain to point where 24 deg. is measured.

Tan24 = h/d, h = d*Tan24.
Tan27 = h/(d-1000), h = (d-1000)Tan27.

d*Tan24 = (d-1000)*Tan27,
d = (d-1000)*1.11,
0.9d = d-1000,
d = 10,000 Ft.
h = d*Tan24 = 4,450 Ft.

Make a sideview sketch

you should have 2 triangles, one right-angled containing the height and a scalene triangle with angles 24° , 153° (the supplement of 27°) and 3°
the side opposite the 3° angle is 1000
by let the side opposite the 24° be x, (also the hypotenuse of the right-angled triangle)

x/sin24 = 1000/sin3
x = 1000sin24/sin3

let the height of the mountain be h
sin 27 = h/x
h = x sin27 = (1000sin24/sin3)(sin27)
= 3528.25