Answer:
The rebate should be $220
Step-by-step explanation:
Demand Curve
It's the relationship between price (P) and quantity (Q) demanded a certain product or service.
(a) We need to find the function that relates both magnitudes assuming a linear equation. The equation of a line can be found with the point-point formula:
[tex]\displaystyle Q-Q_1=\frac{Q_2-Q_1}{P_2-P_1}(P-P_1)[/tex]
Two sets of data are given: 100 Blu-ray disc players are sold a week at $600 each. The ordered pair for this condition is (P,Q)=(600,100).
The other point comes from the market survey: The number of units sold will increase by 80 (100+80=180) when the price goes down $40 (600-40=560). The new point is (P,Q)=(560,180)
We set up the equation of the demand
[tex]\displaystyle Q-100=\frac{180-100}{560-600}(P-560)[/tex]
Rearranging
[tex]-40Q+4000=80P-44800[/tex]
Or
[tex]80P+40Q=48800[/tex]
Simplifying
[tex]2P+Q=1120[/tex]
(b) The revenue function is Q times the price
[tex]R=Q.P[/tex]
Solving the equation of the demand for P
[tex]\displaystyle P=\frac{1120-Q}{2}[/tex]
Thus, the revenue is
[tex]\displaystyle R=Q\cdot \frac{1120-Q}{2}[/tex]
[tex]\displaystyle R=\frac{1120Q-Q^2}{2}[/tex]
(c) To find the optimum value of the revenue, we take the derivative of R and equate to 0
[tex]\displaystyle R'=\frac{1120-2Q}{2}=560-Q=0[/tex]
Solving
Q=560 units a week
For which the revenue is
[tex]\displaystyle R=\frac{1120(560)-560^2}{2}[/tex]
[tex]R=\$156,800[/tex]
And the price is
[tex]\displaystyle P=\frac{1120-560}{2}=280[/tex]
[tex]P=\$280[/tex]
The rebate should be $600-$280=$220
