College Algebra Help Please!?

Question

Use these facts to solve the problem: the radius of the Earth is 3960 miles and one mile equals 5280 feet.
Here's the question:
A person who is six feet tall stands on the beach in Fort Lauderdale, Florida, and looks out onto the Atlantic Ocean. Suddenly, a ship appears on the horizon. How far is the ship from shore?

Can anybody help me solve this? I'm having trouble!

Answer 1

OK, I think I have an idea.

Draw a circle, then draw a vertical line on the left that is tangent to the circle.

Then draw a radius to the point where the line is tangent to the circle.

Then draw another radius at an angle θ (maybe 45°, although thats unrealistic, but just so you can see what's happening) to the radius you just drew...and extend that radius out until it hits the tangent line.

The portion of the line you just drew that is outside the circle is the 6-foot-tall person.

So the length of that line (the radius plus the height of the person) is (3960 miles) x (5280 feet/mile) + 6 feet = 20,908,806 feet.

It's also the hypotenuse of a right triangle...and the base is the radius of the earth (20,908,800 feet).

So the length of the other side can be found using the Pythagorean Theorem:

d² + 20908800² = 20908806²
d² = 250905636
d = 15,840 feet x (1 mile/5,280 feet) = 3 miles

Interesting problem!!!

Answer 2

Assuming the earth is spherical and the surface of the Atlantic Ocean is an unruffled part of that spherical surface and that the person is standing right at the edge of the water and that the ship is a dot, (rather than something standing even taller than six feet)!!, it seems to me we've already involved in so much fancy we can assume the answer is anything we like, although in the 1960s the Four Preps suggested it was 26 miles.

Anyway, if we call the radius r feet (because it's shorter than 3960*5280) and the distance to the ship x feet, that distance is a tangent to the sphere, so it makes a rightangle with the radius, and then we have a rightangled triangle with sides
x, r, r + 6, and so

x^2 = (r+6)^2 - r^2
which =6*(2r + 6)

We'd want the answer in miles, so divide x by 5280.

x = sqrt(6*(2r+6))/5280

I got 3 for that, but I think it's wrong.

OK, I must be right, Jim got the same answer.

Answer 3

Draw a circle. Draw a line extending a little way from it (representing the person) and draw in a line from the other end of that one that just touches the circle (tangent). This represents how far that person can see, i.e. their horizon, which is where the ship is. If the radius of the circle is r and the first line has length h, there is a triangle connecting the end of that line, the centre of the circle, and the point where the tangent line touches.

The distance from the centre of the circle to the tangent point is r. Furthermore, the angle at this point is a right angle. The hypotenuse has length r + h, and we want the length of the other side.

So Pythagoras' theorem give us x = √((r+h)^2 - r^2)
= √(r^2 + 2rh + h^2 - r^2)
= √(2rh + h^2)
= 15840 feet
= 3 miles.
Note that we can ignore the h^2 term in comparison to the 2rh term since h is far smaller than r. In this case it makes a difference of approximately 0.001 foot or 1/73 of an inch (or 1/3 mm in REAL units). Alternatively, it's less than one part in 10 million; we'd have to have all the figures given to 7 decimal places for this to matter.