Question

Helppp meee plssss

Answer 1

Answer:

$a)200x+250y \leq 44000\\ x+y \leq 150,\ where\ x\ is\ the\ number\ of\ children,\ and\ y\ is\ the\ number\ of\ adults.\\ c)The\ number\ of\ children\ and\ adults\ respectively\ maybe\ (20,100),\\ (100,20),(40,40),(80,60).$

Step-by-step explanation:

$We\ are\ given\ that,\\Cost\ of\ the\ Ticket\ To\ the\ Play\ for\ Adults= 250\ php\\Cost\ of\ the\ Ticket\ To\ the\ Play\ for\ Children=200\ php\\Maximum\ number\ of\ Adults\ and\ Children\ permitted\ to\ watch\ the\ play=150\ heads\\Maximum\ amount\ to\ be\ collected\ after\ the\ sales=44,000\ php$

$a)Let\ the\ number\ of\ Children\ who\ attended\ the\ play\ be\ x.\\Let\ the\ number\ of\ Adults\ who\ attended\ the\ play\ be\ y.\\Hence,\\(Total\ Cost\ of\ the\ ticket\ for\ Children)+(Total\ Cost\ of\ the\ ticket\ for\\ Adults) \leq 44,000.\\(No.\ of\ Children\ who\ attended\ the\ play)+(No.\ of\ Adults\ who\ attended\ the\ play) \leq 150.\\Hence,\\200x+250y \leq 44000\\x+y \leq 150$$Hence,\\We've\ found\ the\ inequalities\ that\ best\ represent\ our\ conditions, and\ paired\\ them\ into\ a\ system\ of\ inequalities.$

$b) [Kindly\ refer\ the\ graph\ attachment]\\\\We've\ drawn\ the\ graph\ fixing\ the\ scale\ of\ 1\ unit\ to\ be\ 20\ heads (Adults\ or\ Children).\\The\ red\ slope\ and\ the\ red\ area\ shaded\ below\ it,\ represent\ Inequality\ 1.\\The\ blue\ slope\ and\ the\ blue\ area\ shaded\ below\ it,\ represent\ Inequality\ 2.\\As\ we\ already\ know\ that,\\The\ overlapped\ region\ contains\ the\ solutions\ to\ the\ system\ of\ Inequalities.$

$c)Considering\ any\ four\ points\ on\ the\ overlapped\ region,\ would\ work\ great.\\Also,\\We\ need\ to\ be\ a\ bit\ cautious\ here\ and\ consider\ solutions\ only\ from\ the\\ First\ Quadrant,\ as\ the\ physical\ quantity-Counting\ Numbers\ can\\ only\ be\ Whole\ Numbers.$

$Now,\\Lets\ consider\ points\ (20,100), (100,20),(40,40),(80,60).\\Lets\ verify\ for\ each\ of\ them\ separately:$

$I]Substituting\ x=20,\ y=100\ in\ Inequality\ 1:\\200x+250y \leq 44000\\200*20+250*100 \leq 44000\\4000+25000 \leq 44000\\29000 \leq 44000\\This\ is\ TRUE.\\\\Substituting\ x=20,\ y=100\ in\ Inequality\ 2:\\x+y \leq 150\\20+100 \leq 150\\120 \leq 150\\This\ is\ TRUE\ too.$

$II]\ Substituting\ x=100,\ y=20\ in\ Inequality\ 1:\\200x+250y \leq 44000\\200*100+250*20 \leq 44000\\20000+5000 \leq 44000\\25000 \leq 44000\\This\ is\ TRUE.\\\\Substituting\ x=100,\ y=20\ in\ Inequality\ 2:\\x+y \leq 150\\100+20 \leq 150\\120 \leq 150\\This\ is\ TRUE\ too.$

$III]\ Substituting\ x=40,\ y=40\ in\ Inequality\ 1:\\200x+250y \leq 44000\\200*40+250*40 \leq 44000\\8000+10000 \leq 44000\\18000 \leq 44000\\This\ is\ TRUE.\\\\Substituting\ x=40,\ y=40\ in\ Inequality\ 2:\\x+y \leq 150\\40+40 \leq 150\\80 \leq 150\\This\ is\ TRUE\ too.$

$IV]\ Substituting\ x=80,\ y=60\ in\ Inequality\ 1:\\200x+250y \leq 44000\\200*80+250*60 \leq 44000\\16000+15000 \leq 44000\\31000 \leq 44000\\This\ is\ TRUE.\\\\Substituting\ x=80,\ y=60\ in\ Inequality\ 2:\\x+y \leq 150\\80+60 \leq 150\\140 \leq 150\\This\ is\ TRUE\ too.$

$Hence,\ Verified.$