A 90-foot rope from the top of a tree house to the ground forms a 45∘45∘ angle of elevation from the ground. How high is the top of the tree house? Round your answer to the nearest tenth of a foot.
+45 i'm not quite sure i'm interpreting the problem right, but i'm going to solve the problem assuming that the problem is asking you to find the leg length of a 45-45-90 triangle, where one leg is on the ground and another leg is the height of the treehouse, with the 90-foot rope being the hypotenuse.
we know that the hypotenuse of a 45-45-90 triangle with leg length \(x\) is \(x\sqrt2\), so we know that the leg length of a 45-45-90 triangle with hypotenuse \(y\) is \(\frac{y}{\sqrt2} = \frac{y\sqrt2}{2}\), so plugging in \(y = 90\) (since the rope is the hypotenuse), we get
\(\frac{y\sqrt2}{2} = \frac{90\sqrt2}{2} = \boxed{45\sqrt2}.\)
