Your company manufactures two models of speakers, the Ultra Mini and the Big Sta

Question

Your company manufactures two models of speakers, the Ultra Mini and the Big Stack. Demand for each depends partly on the price of the other. If one is expensive, then more people will buy the other. If p1 is the price of the Ultra Mini, and p2 is the price of the Big Stack, demand for the Ultra Mini is given by q1(p1, p2) = 100,000-500p1 + 10p2 where q1 represents the number of Ultra Minis that will be sold in a year. The demand for the Big Stack is given by 20p1 P2) 150,000 + 10p1- 500p Find the prices for the Ultra Mini and the Big Stack that will maximize your total revenue. (Round your answers to two decimal places.) p1 = $ Submit Answer Save Progress

Explanation / Answer

We have given q1(p1,p2)=100,000-500p1+10p2 and q2(p1,p2)=150,000+10p1-500p2

we want maximum revenue or maximum of p1*q1+p2*q2

p1*q1+p2*q2=p1*(100,000-500p1+10p2 )+p2*(150,000+10p1-500p2)

p1*q1+p2*q2=100,000p1-500p12+10p1p2+150,000p2+10p1p2-500p22

p1*q1+p2*q2=100,000p1-500p12+20p1p2+150,000p2-500p22 ---(1)

differentiating p1*q1+p2*q2 with respect to p1 and p2

d(100,000p1-500p12+20p1p2 +150,000p2-500p22)/dp1=100,000-1000p1+20p2 since p2 is constant

d(p1*q1+p2*q2)/dp1=100,000-1000p1+20p2

d(p1*q1+p2*q2)/dp2=d(100,000p1-500p12+20p1p2+150,000p2-500p22)/dp2=20p1+150,000-1000p2

since p1 is constant

d(p1*q1+p2*q2)/dp2=20p1+150,000-1000p2

then set these equal each to each other to solve for one in terms of the other.

d(p1*q1+p2*q2)/dp1=d(p1*q1+p2*q2)/dp2

100,000-1000p1+20p2=20p1+150,000-

20p2+1000p2=20p1+150,000-100,000+1000p1

1020p2=1020p1+50,000

p2=p1+50,000/1020

substitute back p2=p1+50,000/1020 into equation (1)

100,000p1-500p12+20p1(p1+50,000/1020) +150,000(p1+50,000/1020)-500(p1+50,000/1020)2

100,000p1-500p12+20p12+50,000p1/51) +150,000p1+(150,000*50,000)/1020)-500p12-(50,000/1020)2-1000*(50,000/1020)p1

differentiating the above equation with respect to p1 and set equal to 0 to find maximum

100,000-1000p1+40p1+50,000/51 +150,000+0-1000p1-0-1000*(50,000/1020)=0

-2000p1+40p1+250,000+50,000/51-1000*(50,000/1020)=0

1960p1=250,000+(50,000/51)-1000*(50,000/1020)

p1=[250,000+(50,000/51)-1000*(50,000/1020)]/1960

p1=103.041216487

substitute p1 into p2=p1+50,000/1020

p2=103.041216487+(50,000/1020)

p2=152.06082433

the price for the Ultra Mini is p1=$103.04

the price for the Big Stack is p2=$152.06

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