STATEMENT OF PROBLEM

(a) Consider the region bounded above by the function

f(x) = 9 + 8x - x^2

and below by the x-axis.

Determine the largest circle bounded by the region.

(b) Consider the region bounded above by the function

f(x) = - x^4/20 + 11/15 x^3 - 17/5 x^2 + 24/5 x + 6

and below by the x-axis.

Determine the largest circle bounded by the region.

(c) Consider the region bounded above by the function


f(x) = 8 + 6.2 x - 1.7 x^2 + .1 x^3


and below by the x-axis between x = -1 and x = 8.


Determine the largest circle bounded by the region.

While this would appear to be an abstract only problem consider these two situations.

(1) We have purchased a great number of identical scrap plates of sheet metal and we wish to cut out the largest circle bounded by the boundaries of this scrap sheet for making another product.

(2) We have an irregular shaped field and we wish to set up a circular irrigating sprinkler system. How much area can we irrigate, i.e. what is the largest circle we can inscribe in this field, its radius, and its center?