The telephone company is planning to introduce two new types of executive commun
ID: 2859247 • Letter: T
Question
The telephone company is planning to introduce two new types of executive communications systems that it hopes to sell to its largest commercial customers. It is estimated that if the first type of system is priced at x hundred dollars per system and the second type at y hundred dollars per system, approximately 408x+5y |consumers will buy the first type and 50+9x7y| will buy the second type. If the cost of manufacturing the first type is $1200 per system and the cost of manufacturing the second type is $3000 per system, what prices x and y will maximize the telephone company's profit?
First type: x|= hundred dollars per system
Second type: y |= hundred dollars per system
Explanation / Answer
revenue R(x,y)=x( 408x+5y) +y( 50+9x7y)
revenue R(x,y)=40x8x2+5xy + 50y+9xy7y2
revenue R(x,y)=40x8x2+14xy + 50y7y2
cost of manufacturing the first type is 12 hundred $ per system and the cost of manufacturing the second type is $30 hundred per system
cost C(x,y)=12(408x+5y)+30(50+9x7y)
cost C(x,y)=48096x+60y+1500+270x210y
cost C(x,y)=1980+174x150y
profit P(x,y)=revenue R(x,y)-cost C(x,y)
profit P(x,y)=(40x8x2+14xy + 50y7y2)-(1980+174x150y)
profit P(x,y)=40x8x2+14xy + 50y7y2-1980-174x+150y
profit P(x,y)=8x2+14xy 7y2-1980-134x+200y
for critical points Px=0,Py=0
Px =-16x +14y -134=0
=>-8x+7y-67=0
=>-8x+7y=67------>(1)
Py=14x-14y+200=0
y-x =200/14
y=x+(100/7)---->(2)
put (2)in (1)
-8x+7(x+(100/7))=67
-8x +7x +100=67
x =100-67
x =33
y=33+(100/7)
y=331/7
y=47.286
prices x=33 hundred dollars and y=331/7=47.3hundred dollars per system will maximize the telephone company's profit
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