Answer: radius = 108.75 feet
Hi!
In the drawing you can see the arch that goes from point A to B. The right line from A to B is the length L = 150 feet, across the river. The height is h = 30 feet.
There is a right triangle with hypotenuse R, and the legs are (R-h), and L/2. The Pythagorean theorem says that:
[tex]R^2 = (R-h)^2 + (\frac{L}{2}) ^2 = R^2 -2hR + h^2 + (\frac{L}{2}) ^2\\[/tex]
Then:
[tex]0 = -2hR + h^2 +(\frac{L}{2}) ^2\\R = (h^2 +(\frac{L}{2}) ^2)\frac{1}{2h}[/tex]
Plugging the values of L and h, you get R = 108.75 feet
