Traveling across a flat land you notice a mountain of f in the distance with an angle of elevations 3.5 degrees. You walk toward the mountain 13 more miles and now the angle of elevation is 9 degrees. How tall is the mountain?
To find the height of the mountain, we can use trigonometry and the concept of similar triangles.
Let's start by drawing a diagram to better visualize the situation. Draw a flat land with a mountain located at some distance away from you. The initial angle of elevation to the top of the mountain is 3.5 degrees, and after walking 13 more miles towards the mountain, the angle of elevation increases to 9 degrees.
Now, let's label the unknown height of the mountain as "h" and the distance from your initial position to the mountain as "x" miles.
From the diagram and the given information, we have a right triangle formed by the mountain, your initial position, and your final position (after walking 13 miles towards the mountain). The height of the mountain forms the vertical leg of the triangle, and the horizontal leg represents the distance covered by walking towards the mountain.
Using trigonometry, we can write the following equation for the tangent of the angles of elevation:
tan(3.5 degrees) = h / x
tan(9 degrees) = h / (x + 13)
Now, let's solve this system of equations to find the value of "h":
h / x = tan(3.5 degrees) ------ (Equation 1)
h / (x + 13) = tan(9 degrees) ------ (Equation 2)
Multiply Equation 1 by (x + 13) and Equation 2 by x to eliminate h:
h * (x + 13) / x = x * tan(3.5 degrees) ------ (Equation 3)
h / x = (x + 13) * tan(9 degrees) ------ (Equation 4)
Now, cross-multiply Equation 3 and Equation 4:
h * (x + 13) = x^2 * tan(3.5 degrees) ------ (Equation 5)
h * x = (x + 13) * x * tan(9 degrees) ------ (Equation 6)
Divide Equation 5 by x and Equation 6 by (x + 13) to solve for h:
h = x * tan(3.5 degrees) ------ (Equation 7)
h = x * (x + 13) * tan(9 degrees) / (x + 13) ------ (Equation 8)
Since h is the same in both Equation 7 and Equation 8, we can equate them and solve for h:
x * tan(3.5 degrees) = x * (x + 13) * tan(9 degrees) / (x + 13)
Cancel out the common factors and simplify:
tan(3.5 degrees) = tan(9 degrees) * x / (x + 13)
Now, plug in the values for the tangents:
0.06107 = 0.1583 * x / (x + 13)
Cross-multiply and solve for x:
0.06107 * (x + 13) = 0.1583 * x
0.79391 + 0.079891x = 0.1583x
0.079891x - 0.1583x = -0.79391
-0.078409x = -0.79391
x = -0.79391 / -0.078409
x ≈ 10.1156
Now that we have the value of "x", we can find the height of the mountain by substituting it into Equation 7:
h = 10.1156 * tan(3.5 degrees)
h ≈ 0.6142 miles
Therefore, the approximate height of the mountain is 0.6142 miles.
If the height is h miles, then
h cot3.5° - h cot9° = 13