Question

SORU 1 Determine convergence or divergence of the alternating series. Co 6n+2 [(-1)"In 5n+1 n=1 O A Converges O B. Diverges SORU 2 5 points A rectangular box with square base and no top is to have a volume of 32m². What is the least amount of material required? 36m2 OB. 38m² 42m2 OD. 38m2 ОЕ 48m2

Answer 1

Theorem A series {an in convergent then Liman (Must) 0 Given Given series { (igh en 6n+2 nel Int Check Lim (an) lim (-ign In/6n+2 n 100 sna) (som en non nyo 1) nyo Here um tinen (6+) (5+1) as nyoo limfoon en +0 + en ( ( ) lim (an) gives two values of limit so limit doesn't exist limit of sequence is unique) lim #o nto { an i.e - diverges no thus an Hence diverges 2 (-)” 1 ( 6 n+2 5 nt] na) CSScanned with CamScanner

Some basic concept Let Point of minima of fox) Point of maxima of fix) be a function is said to to be r f(x) 2a e x it (1) f'(a) = 0 (11) fica) 70 aza is said it O f'(a) = 0 (11) f"(a) < 0 Now From question side of square base height of rectangular box eh cx volume of rectangular base - Lxbxh x2h 32 x2h = given volume of box = 32 32 ha 32 32 Amount of material surface area) - Total surface area Areo. ob top surface E2(22+ x n + an) 4x4 +22 Let denone arount s = 4ah tua of material is S = 4 Yxx 32 h = 32 72 S- 128 S (3) - 128473 x Now we have find point of minima of function S(x) S'(X) =0 (128+23961) (vul +x2 22 32 2 } +22 2 al3x2) 2x 2x3 -128 => 2x3 =128 a 2 x3 264 2-4 Here S'(2) 2 x 3 -128 22 S'(4)=0 ICS Scanned with CamScanner...

S'(x) = 2x3-128 275,128 s"(x) 22 (622) (2x3-128)(2x) 24 S''(x) 624 4 x 4 + 256x x 4 S"(x) - 2 x 4 + 2562 Put 34 x = S'(4) > 0 Hence 5'(4) = 0 S" (4)70 thus a = 4 is point of minima of sla) so S(x.) - 128 +23 x S(4) a 128 +43 4 S(4) = 192 こ 48 4 so least amount material required = 48 m2 option E is correct CSScanned with CamScanner