Answer:
Explanation:
To find the shortest length of cable needed, we can use trigonometry to calculate the length of the hypotenuse of a right triangle formed by the mountain, the cable car, and the ground.
Let's denote the height of the mountain as H = 3,400 feet and the horizontal distance from the base of the mountain to the point where the cable car will be installed as D = 940 feet.
In the right triangle, the vertical leg represents the height of the mountain, the horizontal leg represents the distance from the base to the point where the cable car is installed, and the hypotenuse represents the length of the cable.
We can use the sine function to relate the angle of inclination, the height, and the hypotenuse of the triangle:
sin(angle) = opposite/hypotenuse
In this case, the angle of inclination is 74 degrees, and the opposite side is the height of the mountain (H).
Therefore, we can write the equation as:
sin(74 degrees) = H/hypotenuse
Rearranging the equation, we have:
hypotenuse = H / sin(74 degrees)
Let's calculate the value of sin(74 degrees) and substitute the values to find the length of the hypotenuse:
sin(74 degrees) ≈ 0.9613
hypotenuse = 3,400 / 0.9613 ≈ 3,535.89 feet
Therefore, the shortest length of cable needed is approximately 3,535.89 feet when rounded to two decimal places.