A ski resort is installing tows on its new ski runs. The baby hill has a vertical height of 500.0 m and is inclined at 42o. Determine the appropriate length of the T-bar cable if it goes to the top of the hill and returns.

Make a sketch and you will see that

sin 42° = 500/length of hill
length of hill = 500/sin42°

continue from there

223

vdv

To determine the length of the T-bar cable, we need to use trigonometry and the given information.

First, let's break down the information provided:

Vertical height (h) of the baby hill = 500.0 m
Incline angle (θ) = 42 degrees

Now, let's use trigonometry to find the length of the T-bar cable.

We can use the sine function to relate the angle and the side lengths of a right triangle:

sin(θ) = opposite/hypotenuse

In this case, the side opposite to the angle θ is the vertical height (h) of the hill, and the hypotenuse represents the length of the T-bar cable.

So, we have:

sin(θ) = h/hypotenuse

Rearranging the equation, we get:

hypotenuse = h/sin(θ)

Let's substitute the values into the equation:

hypotenuse = 500.0 m / sin(42)

Using a scientific calculator, we can determine that sin(42) ≈ 0.6691.

So:

hypotenuse = 500.0 m / 0.6691

Calculating this, we get:

hypotenuse ≈ 747.22 m

Since the T-bar cable goes to the top of the hill and returns, we need to double this length:

T-bar cable length = 2 * 747.22 m ≈ 1,494.44 m

Therefore, the appropriate length of the T-bar cable is approximately 1,494.44 meters.