To solve this problem, we can use the trigonometric functions sine, cosine, and tangent.
First, let's draw a diagram:
A (top of the mountain)
/|
/ |
/ | h (height of mountain)
/ |
/ θ |
B /_____|
d (distance from base)
We are given that the distance from the base of the mountain to point B is 1/2 mile, or 2640 feet (since there are 5280 feet in a mile). We are also given that the angle of elevation from point B to point A is 65 degrees, and that the angle between the ski lift and the ground is 15 degrees.
Let's start by finding the height of the mountain h. We can use the tangent function, since we know the opposite (h) and the adjacent (d) sides of the right triangle formed by points A, B, and the foot of the mountain (call it point C):
tan(65) = h / d
h = d * tan(65)
Plugging in the values, we get:
h = 2640 * tan(65)
h ≈ 7855 feet
Next, let's find the length of the ski lift. We can use the cosine function, since we know the adjacent (d) and hypotenuse (L) sides of the right triangle formed by points B, the foot of the mountain, and the base of the ski lift (call it point D):
cos(15) = d / L
L = d / cos(15)
Plugging in the values, we get:
L = 2640 / cos(15)
L ≈ 2736 feet
Therefore, the length of the ski lift from the beginning to the end is approximately 2736 feet.