Many sailboats are moored at a marina 4.4 km away on the opposite side of a lake

Question

Many sailboats are moored at a marina 4.4 km away on the opposite side of a lake. You stare at one of the sailboats because, when you are lying flat at the water's edge, you can just see its deck but none of the side of the sailboat. You then go to that sailboat on the other side of the lake and measure that the deck is 1.5 m above the level of the water. You are a distance = 4.4 km from the sailboat when you can see only its deck and not its side. Because of the curvature of the Earth, the water "bulges out" between you and the boat. Estimate the radius of the Earth.

So I'm thinking the equation is the Pythagorean theorem a2+ b2 = c2 where a = distance from you to the boat's deck, b = Earth's radius, and c = Earth's radius + height of the boat from the surface of the lake.

So would the equation be (4400 m)2 + b2 = (b + 1.5 m)2 ?

If so, I got the following work:

19360000 + b2 = b2 + 3b +2.25

1935997.75 = 3b

b = 6.5x105 m

This is not correct though... what did I do wrong?

Explanation / Answer

You used the wrong sceintific notation. All of your work is correct, but instead of b=6.5*10^5 it should be 6.5*10^6m.

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