find the maximum height of the arch.

Question

an arch is in the shape of a semicircle. at a point along the base 3 foot from an end of the arch, the height of the arch is 10 feet. Find the maximum height of the arch.

Answer 1

Draw the semicircle with O as center and point A at left end of the horizontal diameter. Go right 3 units to point B, somewhere between points A and O.  At B raise a perpendicular that meets the upper semicircle at point C.

 

Triangle CBO is a right triangle; OC = r (radius)...hypotenuse

Legs:   BC = 10    OB = r-3

 

Using Pythagorean formula:  r² = (r-3)² + 10²

 

Expand the squared binomial correctly (do not forget the middle term) and you will get a linear equation, after simplifying, and if my arithmetic is correct you get r = 109/6 or about 18.17 ft; this radius is in fact the max. height.

 

Check the work!