Respuesta :
(a) 40 unit have to be the number of units sold that will yield maximum revenue.
(b) The price per unit that will yield maximum revenue, will be $1.195.
(a) We have to find the number of units sold that will yield maximum revenue
Given demand function is
D(x)=130e^{-0.025x}
where is the number of units sold each week
Revenue function R(x) = xD(x)
R(x) = x∙130e^{-0.025x}
To find maximum revenue put,
R’(x) = 0
⇒d/dx(x∙130e^{-0.025x}) = 0
⇒130 d/dx(x∙e^{-0.025x}) = 0
⇒130[e^{-0.025x}+xe^{-0.025x} ∙(-0.025x)]=0
⇒130e^{-0.025x} ∙(1-0.025x)=0
⇒1 - 0.025x=0
⇒0.025x = 1
⇒x = 1/0.025
⇒x = 40units
R''(x) = 130d/d[x(e^{-0.025x}-0.025xe^{-0.025x} )]
R''(x) = 130[-0.025e^{-0.025x}-0.025e^{-0.025x}+0.00625xe^{-0.025x}]
R''(x)|x=40 = 130-0.025*40[0.050+0.00625*40]
R''(x)|x=40 = 130/e ∙(-0.025)
R''(x)|x=40 = (130*(-0.025))/e < 0
At x= 40 units, the revenue will maximize.
$0 units will yield maximum revenue.
(2) Now we have to find the price per unit that will yield maximum revenue.
D(x) = 130e-0.025x
D(40) = 130e^{-0.025*40}
D(40) = 130e^{-1}
D(40) = 130/e
D(40) = 47.82(in dollar)
Price per unit = d(40)/40
Price per unit = 47.82/40
Price per unit = $1.195
The price per unit that will yield maximum revenue, will be $1.195.
To learn more about to maximum revenue link is here
brainly.com/question/1332812
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