Almost embarrasingly simple

[captainnofun]

Hi, I'm new to the forum, and loathe to admit that I'm well out of high school and have the mathematical knowledge of a snowpea. Can anyone help me with the following scenario?

A person drops down a 200 ft. vertical shaft. Now underground, they will travel through a tunnel at a downward slope until they reach a depth of 1 and 1/2 miles. My question is, how steep do I need to make my slope? Since they are walking, it would need to be shallow enough that they can comfortably remain upright, but also steep enough that they will reach their target depth within 1 hour, 2 hours if it's more logical. I appreciate any help you guys can give me!

 

[arildno]

For simplicity, let the velocity they travel with be called V, and the time to reach their destination T (in your case 1 or 2 hours).

Thus, the length L of the tunnel must be L=VT
Agreed so far?

Now, as measured from the starting point of the tunnel, the target point lies a depth D beneath it, (D equals 1 and a half miles in your case)

Now, D and L can be seen as two sides in a right-angled triangle, agreed?
(L is the hypotenuse of this triangle!)

The slope is the angle between the horizontal side H of this triangle and L.

In general, if we let $\theta$ denote the slope angle, we have:
$$\theta=\sin^{-1}(\frac{D}{L})=\sin^{-1}(\frac{D}{VT})$$
where $\sin^{-1}()$ is the inverse sine.

As you can see, you must know what the traveling velocity is in order to calculate the slope angle.

 

[tacman]

Hi there! Don't worry about your mathematical knowledge, we all have different areas of expertise and it's great that you're seeking help on this forum.

To answer your question, the slope of the tunnel would need to be approximately 1.5 degrees for the person to reach a depth of 1.5 miles within 1 hour. This can be calculated using the formula for slope, which is rise over run. In this case, the rise would be 1.5 miles (7920 feet) and the run would be 200 feet.

So the slope would be calculated as (7920/200) x 100 = 3.96 degrees. This is slightly steeper than what you were looking for, so if you want to make the slope more gradual, you can adjust the time frame accordingly. For example, if you want the person to reach the target depth in 2 hours, the slope can be reduced to 1.98 degrees.

I hope this helps and good luck with your scenario!