a zipline starts 120 feet above the ground and covers a diagonal distance, forming a triangle with the ground. if the angle of elevation where the zip line meets the ground is 65 derees, what is the horizontal distance from the base of the tower to where the zip lin ends
To find the horizontal distance from the base of the tower to where the zip line ends, we can use trigonometry.
Let x be the horizontal distance.
We can set up a right triangle with the vertical distance (120 ft), the horizontal distance (x), and the angle of elevation (65 degrees).
Using the sine function, we have:
sin(65) = opposite/hypotenuse
In this case, the opposite side is the vertical distance (120 ft) and the hypotenuse is the diagonal distance (which we need to find).
sin(65) = 120 ft / hypotenuse
Multiplying both sides by the hypotenuse, we get:
hypotenuse = 120 ft / sin(65)
Using a calculator, we can find the value of sin(65) to be approximately 0.9063.
Therefore, the hypotenuse is approximately:
hypotenuse = 120 ft / 0.9063 ~= 132.17 ft
Finally, the horizontal distance x is the adjacent side of the triangle.
Using the cosine function, we have:
cos(65) = adjacent/hypotenuse
cos(65) = x / 132.17 ft
Multiplying both sides by 132.17 ft, we get:
x = 132.17 ft * cos(65)
Using a calculator, we can find the value of cos(65) to be approximately 0.4226.
Therefore, the horizontal distance x is approximately:
x = 132.17 ft * 0.4226 ~= 55.91 ft
So, the horizontal distance from the base of the tower to where the zip line ends is approximately 55.91 feet.