Respuesta :
Answer:
1). [tex]P'(t) = (-9025t).e^{-0.05(53+0.95t^2)}[/tex]
2). (-435.36) dollars per week
Step-by-step explanation:
Weekly price decay of the product is represented by the function,
P(x) = [tex]95000.e^{-0.05x}[/tex]
And the price of the product changes over the period of 't' weeks is represented by,
x(t) = [tex]53+0.95t^2[/tex]
Function representing the rate of change in the profit with respect to the time will be represented by,
1). P'(t) = [tex]\frac{dP}{dx}.\frac{dx}{dt}[/tex]
Since, P(x) = [tex]95000.e^{-0.05x}[/tex]
P'(x) = [tex]95000\times (-0.05).e^{-0.05x}[/tex]
= [tex](-4750).e^{-0.05x}[/tex]
Since, x(t) = 53 + 0.95t²
x'(t) = 1.9t
[tex]\frac{dP}{dx}.\frac{dx}{dt}=(-4750).e^{-0.05x}\times (1.9t)[/tex]
By substituting x = 53 + 0.95t²
[tex]\frac{dP}{dx}.\frac{dx}{dt}=(-4750).e^{-0.05(53+0.95t^2)}\times (1.9t)[/tex]
P'(t) = [tex](-9025t).e^{-0.05(53+0.95t^2)}[/tex]
2). For t = 7 weeks,
P'(7) = [tex](-9025\times 7).e^{-0.05(53+0.95(7)^2)}[/tex]
= [tex](-63175).e^{-4.9775}[/tex]
= (-63175)(0.006891)
= (-435.356) dollars per week
≈ (-435.36) dollars per week
