Answer:
Price = $4.50
Maximum revenue = $2,430
Step-by-step explanation:
The concession stand sells 600 bags of peanut at $4.00 (600, $4.00).
If there is a $1 increase in price, the number of bags old will decrease by 120, which mens that 480 bags of peanuts will be sold at $5.00 (480, $5.00).
Tracing a linear relationship between price and quantity sold with the two given points:
[tex]m = \frac{5-4}{480-600}\\ m=-\frac{1}{120}[/tex]
[tex]P - P_0=m*(Q-Q_0)\\P-4.00=-\frac{1}{120} (Q-600)\\P= -\frac{1}{120}Q +9[/tex]
The revenue function is given by the price multiplied by the quantity sold:
[tex]R=Q*P= (-\frac{1}{120}Q +9)*Q\\R=-\frac{1}{120}Q^2 +9Q[/tex]
The value of 'Q' for which the derivate of the revenue function is zero, is the output level for which revenue is maximum:
[tex]R'=0=-\frac{2}{120}Q+9\\Q_{max} = 540\\P_{max} = -\frac{540}{120} +9\\P_{max} =\$4.50[/tex]
The total revenue of 540 units at $4.50 per unit is:
[tex]R = \$4.50*540\\R=\$2,430[/tex]
