Park

In the newly built park will be permanently placed a rotating sprayer irrigation of lawns. Determine the largest radius of the circle which can irrigate by sprayer P so not to spray park visitors on line AB. Distance AB = 55 m, AP = 36 m and BP = 28 m.

Result

r =  16.19 m

Solution:

$$ \smash{ a = 55 \\~\\b = 36 \\~\\c = 28 \\~\\ \\~\\s = (a+b+c)/2 = 59.5 \\~\\r = = ? \\~\\ \\~\\T = \frac{a r}{2} = \sqrt{s(s-a)(s-b)(s-c)} \\~\\\frac{a r}{2} = \sqrt{s(s-a)(s-b)(s-c)} \\~\\r = \frac{ 2\sqrt{s(s-a)(s-b)(s-c)}}{a} \\~\\r = \frac{ 2 \cdot \sqrt{ 59.5(59.5-55)(59.5-36)(59.5-28)}}{ 55} \\~\\r = 16.19 \ { m } } $$


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