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.
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, [tex]h^2+ 15^2 = (2h +1)^2[/tex]
Simplifying it down, we get [tex]3h^2+4h-224=0[/tex]
We can use the quadratic formula to solve this, where the solutions are [tex] \frac{-4 \pm \sqrt{4^2 - 4 * 3 * (-224)} }{2*3} = \frac{-4 \pm \sqrt{2704} }{6} = \frac{-4 \pm 52}{6} [/tex]
Therefore the answers are [tex]- \frac{56}{6} [/tex] or [tex] \frac{48}{6} =8[/tex]
The answer is 8since the height of the tree must be positive.
