Question

A steep mountain is inclined 74 degree to the horizontal and rises 3400 ft above the surrounding plain. A cable car is to be installed from a point 970 ft from the base to the top of the mountain. Find the shortest length of cable needed.

Answer 1

Using trigonometry, the shortest length of cable needed for the cable car to reach the top of a mountain inclined at 74 degrees and 3400 feet above the plain, with a horizontal distance of 970 feet to the base, is approximately 3637.79 feet.

Step-by-step explanation:

To calculate the shortest length of cable for the cable car, we need to use trigonometric relationships. Given the steep mountain is inclined at 74 degrees and the height above the plain is 3400 feet, and a horizontal distance from the base is 970 feet, we can use the Pythagorean theorem on the resulting right triangle to find the length of the hypotenuse, which represents the shortest length of cable needed.

Let's denote the length of the cable as c. Using the trigonometric relationship:
cos(74 degrees) = 970/c
we can isolate c and calculate it by:
c = 970 / cos(74 degrees).

After calculating we get:

c ≈ 3637.79 feet

Therefore, the shortest length of cable that is needed is approximately 3637.79 feet.

Answer 2

3917m

Step-by-step explanation:

We are given that

`\theta=74^(\circ)`

`h=3400 ft`

We have to find the shortest length of cable needed.

Let Base=x

We know that

`tan\theta=(Perpendicular\;side)/(Base)`

Using the formula

`tan74^(\circ)=(3400)/(x)`

`x=(3400)/(tan74)=975m`

`y=x+970=975+970=1945m`

Now, the length of cable needed

=
`√((3400)^2+(1945)^2)`

=3917m

Hence, the shortest length of cable needed=3917m