Using plant to find depth of water (triangles)

Question

John and Chris were out in their row boat one day, and Chris spied a water lily. Knowing that Pat liked a mathematical challenge, Chris announced that, with the help of the plant, it was possible to calculate the depth of the water under the boat. When pulled taut, the top of the plant was originally 10 inches above the water surface. While Pat held the top of the plant, which remained rooted to the lake bottom, Chris gently rode the boat 5 feet. This forced Pat's hand to the water surface. Use this information to calculate the depth of the water.

I've tried solving for a lot of triangles, but I always end up missing some lengths. I'm guessing you need to use right triangles. I made a right triangle with side lengths 10, 60, and 10 sqrt 37, which basically gave me all the lengths above the water. I'm not sure how to proceed and find the depth of the water,

Answer 1

If $x$ is the depth of the water, then the length of the plant is $\frac{5}{6}+x$ (since $10" = \frac{5}{6}$ feet). So when the boat is rowed $5$ feet away, you get a right triangle with legs $5$ and $x$, and hypotenuse $\frac{5}{6}+x$. I suspect you can take it from there.

Answer 2

Note that the height (length) of the tree doesn't change; there are two triangles to look at.

First, to describe the small right triangle at or above the water level, which we can visualize: we have a leg of $10'' = \frac 56$ft, and a leg of length $5$ft. The small triangle's hypotenuse, above water level, is therefore given by $$h = \sqrt{25 + \frac {25}{36}} = 5\sqrt {1 + \frac 1{36}}ft.$$

This hypotenuse is also the length of the base of an isosceles triangle, whose equal sides are formed by the upright tree (before movement), and the "leaning" tree after the boat rows five feet. Can you take it from here?

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Alternatively, and more easily, we can look at the right-triangle with right-angle formed by legs of length $x$ft (length of the tree below water at the start, i.e. water depth), and $5$ft, whose hypotenuse is the height of the tree (leaning tree after boat's movement): $x + 10''$ft. Now, solve for $x$ using the Pythagorean Theorem:

$$x^2 + 5^2 = (x+\frac 56)^2$$

(Use the only root that makes sense!)