Question

Using derivatives pls. Greatly appreciated!

is going to be hemical storage tank with a capacity of 1000 m constructed in a warehouse that is 12 m by 15 m, with a specifications call for the base to be made of sheet steel house that is 12 m by 15 m, with a height of 11 m. The all for the base to be made of sheet steel that costs $100/m², the top to et steel that costs $50/m?, and the wall to be made of sheet steel that be made of sheet steel that costs $50/mʻ, and the costs $80/m² (a) Determine whether it is possible for a tank of this capacity to fit in the wa is possible, state the restrictions on the radius. (b) If fitting the tank in the warehouse is possible, determine the proportions that meet the conditions and that minimize the cost of the steel for construction. All calculations should be accurate to two decimal places.

Answer 1

@ Dimensions of ware house are 12m by 1sm with Height olm (a) Volume of ware house = 12X15X11 = 1980 mg] i while volume of cylindrical tank = 180oma ne cylinder = HR2H let maximum height of cylinder be Max=11m 9 100o TIRPX !! 9. PRE 1000 R= love = 28.9372624 | ХОТ ERE PF-93 72624 = 5.38m We see that 1000= 1R2H = R² = love TH - When it is maximum =) Radily R is minimum. So cylinder must have minimum radius Rmin=5.38m to Rading of cylinder) R7, 5:38M] but we again have to lind max value of pedity.

We know that dimensions of Base of ware house are 12m and Ism. RM9X 12 15. - It is theorem, that biggest circle inside a rectangle must have its diameter equal to Inalles dimension of Rectangle

Therefore Omax=12m = 2 &max=12 & Rmax=6 So maximum readily of circle Rmax com This tank is possible to lit in the hot warehouse . where min reeding Rush = 5.38m, max. Radling Rmax=6m 15.38 ER SO restriction of radily

6 we have already Calcalted that Minimum Seeding of cylinder is amin=5.38m when height is max Hax=11m Maximum lading of cylinelu Rmax=6 when height is min 100u. tt Rmax remin ) Hmin= 1000 Celooo I Hmin = 8.84m 1. TX 36 3615 Thally [ 5.28 = R 3.38 = Reo = 6 .. 8084 E Hell - Cost of cylindrical Baje = $100/42 Total cost of cylindrical Best= 100XTA2 =100 R? Cost of top= SOX HP2 = SOHR? Total Cost of wall= 80X LARH = 168118h Overall cost (C)= 1004742+ sorp?4 169 pang C = Cootr?+ June 24 160 17ph

430 seә4 рухч е Н= (очь TUR? So Cost function C(R) = 150 HR2+ ZHX 160 HA XClone) TiRt) CCR) = (somme?+ 169.00 Fee miminen Cost, sector and de de Belf OTR fotosco 3 3 HR=1600 - R3 = 160 Therefore t= (160)3 = (169.765273) } ITR= 1600 - R3 = 16oo R2 3 311 R = 5.537 R = 5.54 m) a HE 1000 = 10037 m PT (5554)2 So Specifications for mininum cost (RES.sum 1H=10.37m