From one point on the plain, they observe that the angle of elevation to the top of the mountain is
27
∘
. From a point 1500 feet closer to the mountain along the plain, they find that the angle of elevation is
30
∘
.
How high (in feet) is the mountain?
Draw the diagram, then review your basic trig functions. You will see that the height h can be found using
h cot27° - h cot30° = 1500
so, h = 1500/(cot27° - cot30°)
To solve this problem, we can use the trigonometric relationship between angles of elevation and the height of an object.
Let's break down the information given:
- From one point on the plain, the angle of elevation to the top of the mountain is 27 degrees.
- From a point 1500 feet closer to the mountain along the plain, the angle of elevation is 30 degrees.
Let's denote the height of the mountain as h.
Now let's form a right triangle to represent the situation, where the mountain is the vertical leg, the horizontal distance is 1500 feet, and the line of sight to the top of the mountain from the first observation point is the hypotenuse.
Using the trigonometric relationship, we can write the following equation:
tan(27 degrees) = h / 1500
Now, let's solve for h by multiplying both sides of the equation by 1500:
1500 * tan(27 degrees) = h
Using a calculator, we can find that tan(27 degrees) is approximately 0.5095.
Therefore:
h = 1500 * 0.5095
h ≈ 764.25 feet
Hence, the height of the mountain is approximately 764.25 feet.
To find the height of the mountain, we can use trigonometry and set up a right triangle.
Let's denote the height of the mountain as h and the distance from the original point to the mountain as x.
In the first scenario, we have a right triangle with the opposite side (h) and the adjacent side (x). The angle opposite the height of the mountain is 27°.
In the second scenario, when we move 1500 feet closer to the mountain, we have a right triangle with the opposite side (h) and the adjacent side (x - 1500). The angle opposite the height of the mountain is 30°.
Now, we can use the tangent function to set up the following equations:
For the first scenario:
tan(27°) = h/x
For the second scenario:
tan(30°) = h/(x - 1500)
We need to solve these equations to find the values of h and x.
First, let's solve for x in the first equation:
x = h/tan(27°)
Then substitute this value of x into the second equation:
tan(30°) = h/(h/tan(27°) - 1500)
Now we can solve for h by simplifying the equation.
tan(30°) = h/(h/tan(27°) - 1500)
tan(30°) = (h * tan(27°))/(h - 1500 * tan(27°))
Cross-multiplying:
tan(30°) * (h - 1500 * tan(27°)) = h * tan(27°)
Expanding:
h * tan(30°) - 1500 * tan^2(27°) = h * tan(27°)
Grouping common terms:
h * (tan(30°) - tan(27°)) = 1500 * tan^2(27°)
Dividing both sides by (tan(30°) - tan(27°)):
h = (1500 * tan^2(27°))/(tan(30°) - tan(27°))
Finally, calculate the value of h using the above formula on a calculator, which gives approximately 1039.77 feet.
Therefore, the height of the mountain is approximately 1039.77 feet.