An adjustable water sprinkler that sprays water in a circular pattern is placed at the center of a square field whose area is 1250 square feet. what is the shortest radius setting that can be used if get field is to be completely enclosed within the circle ?

The area of a square is given by s^2, the length of one of its sides squared, meaning its sides must each be 25x sqrt(2) ft long. the longest distance on a square is either of its diagonals, both of which pass through the center, so the longest distance form the center of the field would be equal to half of the diagonal.
the triangle formed by the diagonal and two side of the square would be a 45-45-90 special triangle, meaning the ratio of the sides is 1-1-sqrt(2), so if the the sides of the square are 25 sqrt(2), the diagonal must be 50 ft. this means half of the diagonal must be 25 ft, which is the distance from the sprinkler at the center to the corner of the square field, and the shortest possible radius is 25 ft.