Being some distance from a mountain a man looks up at a 9 degree angle to see the top of the moutain. he then moves 13 miles away from the mountains at looks up at a 3.5 degree angle to see the top of the mountain. How tall is the mountain?
Calculus? This is a standard trig question.
draw the figure, two right triangles.
inner triangle: tan9=h/x x is the distance to the mountan.
outer triangle: tan3.5=h/(x+13)
two equations, two unknowns.
x=h/tan9
x=h/tan3.5 -13
h/tan9=h/tan3.5 - 13
solve for h
To find the height of the mountain, we can use trigonometry. Let's break down the problem into two triangles.
In the first triangle, the man is looking up at a 9-degree angle to see the top of the mountain from a certain distance. Let's assume the height of the mountain is "h" and the distance between the man and the mountain is "x."
In this triangle, we have the opposite side (the height of the mountain) and the adjacent side (the distance from the man to the mountain). Since tangent is defined as the opposite side divided by the adjacent side, we can write the equation:
tan(9 degrees) = h / x
Similarly, in the second triangle, when the man moves 13 miles away from the mountain and looks up at a 3.5-degree angle, we have:
tan(3.5 degrees) = h / (x + 13)
Now, we can solve these two equations to find the height of the mountain.
First, let's find the values of tan(9 degrees) and tan(3.5 degrees):
tan(9 degrees) ≈ 0.1584
tan(3.5 degrees) ≈ 0.0611
Now, let's solve the equations:
0.1584 = h / x (Equation 1)
0.0611 = h / (x + 13) (Equation 2)
From Equation 1, we can rewrite it as h = 0.1584x and substitute it into Equation 2:
0.0611 = (0.1584x) / (x + 13)
Now, we can solve for x:
0.0611(x + 13) = 0.1584x
0.0611x + 0.7953 = 0.1584x
0.1584x - 0.0611x = 0.7953
0.0973x = 0.7953
x ≈ 8.175 miles
Now that we have the value of x, we can substitute it back into Equation 1 to find the height of the mountain:
h = 0.1584x ≈ 0.1584 * 8.175 ≈ 1.295 miles
Therefore, the height of the mountain is approximately 1.295 miles.