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1.7 MODELING WITH FUNCTIONS Who was the roundest knight at King Arthur's Round Table? Sir Cumference From Words to Expressions A number increased by 2 then cut in half 5 decreased by a number then tripled A number decreased by 7 then doubled Area of a Circle We know…. Area=πr² But…. What happens when we have the circumference and we have to find area??? Well…. Area of Circle Cont. C= Circumference Since C= 2πr we can solve for r to get r= C/(2π). Then we can substitute to get area: A= πr²= π(C/2 π))²= πC²/(4π²)= C²/(4π) So … Area= C²/(4π) Example: C= 8 so 8²/(4π)= 5.093 Box Problem A square of side X is cut out of each corner of an 8 in. by 15 in. piece of cardboard and the sides are folded up to form an open-topped box. How big should the cut-out squares be in order to produce the box of maximum volume? 8 x x 15 Solution Volume = Length x Width x Height V = (15-2x) (8-2x) (x) X = 1.667 inches Box Problem 2 A square side is cut out of each corner from a 20cm by 8cm piece of cardboard to form an open-top box. Find the value of x for the box to have the maximum amount of volume. Solution Volume = Length x Width x Height V = (8-2x) (20-2x) (x) X = 1.761 centimeters Box Problem 3 Find the maximum volume. 10 X X 47 Solution Volume = Length x Width x Height V = (10-2x) (47-2x) (x) V = 526.847 units