Question

11 please

11. [10pts.] Find the dimension of the rectangle of the largest area that can be inscribed in a circle of radius r. Find critical number(s) and apply the second derivative test.

Answer 1

N By diagram using pythagoras x + 4y = re ye = r2_x²=y= Jr2_x2 of euctangle Or 24 will be The area A= falley) -29- dx 8x² 2 = 2x125 re-dez) = ya 5r2_x2 If we take the derivative with respect to x, dA = A' = 4 Jr2_x² + 4x(-2x) 25r2_x = 4 502_x2 25r2_x2 = 4 Jr²x2 .4 x² Jr2_x² 410² K² - 4x4 Sp2_m2 4r²_802 © Jp²n Critical numbers :- A'RO 0 = 4,2_8u² Jr²ne 4128x1²=0 AI gre = 8 nez x=fara þ ñ q na x² olos -for-th untur tým Since & mulet be positive 1 then a= lr. 22

(take tuel value 1-20) The value of y will be given by y=512_(272 Szr2= 금 differentiate eg ① again with despect to x, weget (11342)(-1604) -(482-86?) As (Wn2_x272 AC ( 12-0²) (-16x) -(41²-82.) (-x) (5r2_x2)3 A" 1² 1² A" ( 36 ) ก albur2 + 168² +Yare 28x? r2_x23 AT) - 8x² - 12ar2 (m2_x73/2 a”() - Slika 12 metain) 8x In (m2-tre) 3/2 desen - lexandra Å neja elurs 2 1212) r3 -16 Co Since A" (a) co for a = the All) has local maximum at u= lr 4 This the shape will be equare of dimensions th bly For gening maximium of the an J2 area