Question

8. A piece of wire of length L cm is cut into two pieces. One piece, of length x cm, is made into a circle; the rest is made into a square. Denote by A(x) the sum of the area of the circle and the area of the square. (a) (5 points) Find I'm which makes A(z) MINIMUM, and find A(1m). (b) (5 points) Find xm which makes A(x) MAXIMUM, and find A(zm).

Answer 1

wire of length l ut into two pieces se length & Lx cm radius of circle is a as arox llength of wire wire] arcle made of he used t-x em in making square ocx ce (. side of square A) 4 A(x)= sum of area of circle = x2 (2-x² t 4T 16 to find minimum value of Alx) A'(= 22 41 16 let, ACX)=0 23 211-4)= 0 3 8x = 211-x) 16 2 (L-(6) + 2) 40 x = asosnak 4(0) = 12 A (L) = 22 41 AC) = (45)74475) 16 41 2 1612 + 2 tom = 1 400 161² 100 IT > AlXm) = 5 + look tool

A(x) occors maximum value at X=L 1xmax=L & A(X) = 2² 41 ie. Don't cut wise direct make square out of it