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.