Question

A steep mountain is inclined 74 degree to the horizontal and rises to a height of 3400 ft above the surrounding plain. A cable car is to be installed running to the top of the mountain from a point 890 ft out in the plain from the base of the mountain. Find the shortest length of cable needed.

Answer 1

To find the shortest length of cable needed, we can calculate the hypotenuse of the right triangle formed by the mountain slope and the horizontal distance from the base of the mountain to the point where the cable car starts. Using the trigonometric function sine, we can calculate the length of the side opposite the angle, which represents the height of the mountain. Then, using the Pythagorean theorem, we can calculate the hypotenuse, which represents the shortest length of cable needed.

Step-by-step explanation:

To find the shortest length of cable needed, we can calculate the hypotenuse of the right triangle formed by the mountain slope and the horizontal distance from the base of the mountain to the point where the cable car starts.

Using the trigonometric function sine, we can calculate the length of the side opposite the angle, which represents the height of the mountain:

Height = 3400 ft

Then, we can calculate the length of the side adjacent to the angle, which represents the horizontal distance:

Horizontal distance = 890 ft

Finally, using the Pythagorean theorem, we can calculate the hypotenuse, which represents the shortest length of cable needed:

Hypotenuse = √(Height² + Horizontal distance²)

Plugging in the values, we get:

Hypotenuse = √(3400² + 890²) ft

Hypotenuse ≈ 3547.27 ft

Therefore, the shortest length of cable needed is approximately 3547.27 ft.

Answer 2

about 3878 ft

Step-by-step explanation:

Assuming that the cable is not doubled, we need to find the length of the base of the mountain i.e

3400/tan(74) = about 975 ft

Therefore, the length of the cable =

√[ 3400² + (975 + 890)²] = about 3878 ft