A manufacture has been selling 1450 television sets a week at $390 each. A marke
ID: 2878188 • Letter: A
Question
A manufacture has been selling 1450 television sets a week at $390 each. A market survey indicates that for each $22 rebate offered to a buyer, the number of sets sold will increase by 220 per week.
a)Find the function representing the demand p(x), where x is the number of the television sets sold per week and p(x) is the corresponding price.
Answer: 535-x/10
b) How large rebate should the company offer to a buyer, in order to maximize its revenue? (In dollars)
c) If the weekly cost function is 94250+130x, how should it set the size of the rebate to maximize its profit?
Answer: 57.5 dollars
I was able to figure out part a and c but still need help with b. Thanks
Explanation / Answer
a)1450 television sets a week at $390 each
(x1,p1)=(1450,390)
for each $22 rebate offered to a buyer, the number of sets sold will increase by 220 per week.
let rebate =22 dollars
price =390-22=368
sets sold =1450+220 =1670
(x2,p2)=(1670,368)
p-p1=[(p2-p1)/(x2-x1)](x-x1)
p-390=[(368-390)/(1670-1450)](x-1450)
p-390=(-1/10)x+145
p=-(1/10)x+ 535
b)revenue R=p*x
R=(-(1/10)x+ 535)*x
R=(-(1/10)x2+ 535x)
for maximum revenue dR/dx =0,d2R/dx2 <0
dR/dx =(-(1/5)x+ 535),d2R/dx2 =(-1/5)<0
-(1/5)x+ 535=0
(1/5)x= 535
x=2675
revenue is maximum when x=2675
p=-(1/10)x+ 535
p=-(1/10)2675+ 535
p=-267.5+535
p=267.5
rebate should the company offer to a buyer=390-267.5
rebate should the company offer to a buyer=122.5 dollars
c)R=(-(1/10)x2+ 535x)
cost C=94250 +130x
profit P =R-C
P=(-(1/10)x2+ 535x)-(94250 +130x)
P=-(1/10)x2+ 405x-94250
for maximum profit dP/dx=0,d2P/dx2 <0
dP/dx=-(1/5)x+ 405,d2P/dx2 =-1/5<0
-(1/5)x+ 405=0
x=2025
p=-(1/10)x+ 535
p=-(1/10)2025+ 535
p=-202.5+535
p=332.5
rebate should the company offer to a buyer=390-332.5
rebate should the company offer to a buyer=57.5 dollars
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