Question

at a point on the ground 15 ft from the tree, the distance to the top of the tree is 1 ft more than 2 times the height of the tree. find the height of the tree.

Answer 1

Let's say the tree is of height h. The point 15 ft away, the base of the tree, and the height of the tree form a right triangle, where the length of the sides are h, 2h + 1, and 15 (see attached image).

Using the Pythagorean theorem, $h^2+ 15^2 = (2h +1)^2$

Simplifying it down, we get $3h^2+4h-224=0$

We can use the quadratic formula to solve this, where the solutions are $\frac{-4 \pm \sqrt{4^2 - 4 * 3 * (-224)} }{2*3} = \frac{-4 \pm \sqrt{2704} }{6} = \frac{-4 \pm 52}{6}$

Therefore the answers are $- \frac{56}{6}$ or $\frac{48}{6} =8$

The answer is 8 since the height of the tree must be positive.