A product is introduced to the market. The weekly profit (in dollars) of that product decays exponentially as function of the price that is charged (in dollars) and is given by P ( x ) = 95000 ⋅ e − 0.05 ⋅ x Suppose the price in dollars of that product, x ( t ) , changes over time t (in weeks) as given by x ( t ) = 53 + 0.95 ⋅ t 2 Find the rate that profit changes as a function of time, P ' ( t ) dollars/week How fast is profit changing with respect to time 7 weeks after the introduction. dollars/week

1).

2). (-435.36) dollars per week

Step-by-step explanation:

Weekly price decay of the product is represented by the function,

P(x) =

And the price of the product changes over the period of 't' weeks is represented by,

x(t) =

Function representing the rate of change in the profit with respect to the time will be represented by,

1). P'(t) =

Since, P(x) =

P'(x) =

       =

Since, x(t) = 53 + 0.95t²

x'(t) = 1.9t

By substituting x = 53 + 0.95t²

   P'(t) =

2). For t = 7 weeks,

P'(7) =

       =

       = (-63175)(0.006891)

       = (-435.356) dollars per week

       ≈ (-435.36) dollars per week

$30.32

step-by-step explanation:

65.00*.12=2.80

65.00-2.80=57.20

57.20*1.06=60.632

60.632/2=30.316