Word Problem

[Hockeyman]

Ok having some trouble with this word problem, i don't want you guys to tell me what the answer is because i want to figure it out on my own but mabe if you could give me a clue? Here it is:

A man is standing on a boat at the south side of a lake. Assume that the man's feet and the boat are at water level. The man is 5'10" tall. For this problem, say the radius of the Earth is about 3960 miles. The lake is about 23 miles long and assume it is not curved. If the man is at the south end of the lake, can the man see the north end? How far can he see?

Again i don't want the answer but a clue would be great.

 

[skeeter]

Line of sight is tangent to the earth

distance to the point of tangency from the earth's center = earth's radius

distance from the man's eyes to the center of the earth = earth's radius + man's height

sketch all three of the above segments, you'll get a right triangle.

Using Pythagoras ...

L² + R² = (R + h)²

remember, all units for length have to be the same.

 

[soroban]

Hello, Hockeyman!

A man is standing on a boat at the south side of a lake.
Assume that the man's feet and the boat are at water level. The man is 5'10" tall.
For this problem, say the radius of the Earth is about 3960 miles.
The lake is about 23 miles long and assume it is not curved.
If the man is at the south end of the lake, can the man see the north end?
How far can he see?

The units are horrible!
We're dealing with thousands of miles and and a man measured in feet and inches.

                A
                *
                |\
                | \
               h|  \
                |   \L
                |    \
              * * *   \
          *     |     *\
        *       |       *B
       *       R|     /  *
                |   /R
      *         | /       *
      *         *         *
      *         O         *

       *                 *
        *               *
          *           *
              * * *

I see that skeeter already explained the solution . . . very nicely.
I'll modify my diagram to match his explanation.

The man's eyes are at A: \(\displaystyle \,h\,=\,5\frac{5}{6}\) ft.

The radius of the earth is: \(\displaystyle \,R\,=\,3960\) miles \(\displaystyle =\,20,908,800\) ft.

The man's line-of-sight is: \(\displaystyle \,L\,=\,AB\)

Since \(\displaystyle \angle B\,=\,90^o\), we can use Pythagorus to find \(\displaystyle L.\)

 

[Hockeyman]

I got an extremely wierd answer 2.187 x 10^10. Is this right? And how do you know that his line of sight is a tangent?

 

[Gene]

That is wierd.
If the man is h feet high he is H=h/5280 miles high. The triangle has an hypot of R+H, one leg is R and the other is L
R²+L²=(R+H)²=R²+2RH+H²
L²=2RH+H²
H is so much smaller than R that H² can be ignored so
L=sqrt(2R(h/5280)) miles
A little less than 3 miles

The line of sight is a tangent 'cause if he looks higher he sees sky, lower he sees dirt so the horizon is a tangent.

But the problem says the lake is flat, not curved so the whole discussion is moot. The radius of the earth is irrelevant. On a flat surface he can see any distance.