At an altitude of 2000 feet, Ace is flying toward a mountain (a vertical rock face). If the angle of elevation from the plane to the top of the mountain is 20o and the angle of depression to the base of the mountain is 6o, then how high is the mountain (to the nearest foot)? Type only the numerical value.
I have no idea how to even start this question. Any help would be greatly appreciated :)
2000 ft down the mountain is the 6º angle
so his distance to the mountain is
... d = 2000 ft / tan(6º)
the mountain above 2000 ft is
... h = d * tan(20º) = 2000 * tan(20º) / tan(6º)
To solve this problem, you can use trigonometry. Let's label the unknown height of the mountain as 'h'.
From the information given, we can form a right-angled triangle. The plane represents the horizontal side, the mountain represents the vertical side, and the line of sight from the plane to the top and bottom of the mountain represents the hypotenuse.
Using the concept of trigonometry, we can use the tangent function to relate the angles and sides of the triangle:
1. The angle of elevation to the top of the mountain is 20 degrees. This means that the tangent of 20 degrees is equal to the height of the mountain divided by the horizontal distance.
tan(20°) = h / x (Equation 1)
2. The angle of depression to the base of the mountain is 6 degrees. This means that the tangent of 6 degrees is equal to the height of the mountain divided by the horizontal distance plus the height of the plane.
tan(6°) = h / (x + 2000) (Equation 2)
Now, we have a system of two equations with two variables (h and x). We can solve for 'h' by isolating it in one of the equations and substituting it into the other equation.
1. Rearrange Equation 1 to isolate 'h':
h = x * tan(20°)
2. Substitute this value of 'h' into Equation 2:
tan(6°) = (x * tan(20°)) / (x + 2000)
Now, we can solve this equation to find the value of 'x', which represents the horizontal distance from the plane to the base of the mountain.
1. Multiply both sides of Equation 2 by (x + 2000):
(x + 2000) * tan(6°) = x * tan(20°)
2. Distribute on the left side:
(x * tan(6°)) + (2000 * tan(6°)) = x * tan(20°)
3. Move all terms with 'x' to one side and the constant terms to the other side:
(x * tan(6°)) - (x * tan(20°)) = - (2000 * tan(6°))
4. Factor out 'x':
x * (tan(6°) - tan(20°)) = - (2000 * tan(6°))
5. Divide both sides by (tan(6°) - tan(20°)):
x = - (2000 * tan(6°)) / (tan(6°) - tan(20°))
By substituting the values of tan(6°) and tan(20°) into this equation, you can find the value of 'x'. Then, substitute the calculated value of 'x' into Equation 1 to find the value of 'h', which represents the height of the mountain.