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Answer :
The rate of change of weekly sales with respect to time when t = 0 is approximately 2207 units per month.
Hence, the correct option is C.
To find the rate of change of weekly sales with respect to time when t = 0, we need to compute the derivative of the weekly sales function S with respect to t and evaluate it at t = 0.
Given that x = 300t + 2000, we can express x in terms of t
x = 300t + 2000
Now, substitute this value of x into the sales function
S = 60000 - 40000[tex]e^{(-0.0005(300t + 2000)}[/tex]
Simplifying further
S = 60000 - 40000 [tex]e^{(-0.15t - 1)}[/tex]
To find the rate of change of S with respect to t, we differentiate the sales function S with respect to t
dS/dt = -0.15 * (-40000) * [tex]e^{(-0.15t - 1)}[/tex]
Simplifying further
dS/dt = 6000[tex]e^}(-0.15(0) - 1)}[/tex]
Now, we can evaluate the derivative at t = 0
dS/dt = 6000[tex]e^}(-0.15(0) - 1)}[/tex]
= 6000[tex]e^{-1}[/tex]
≈ 6000 * 0.3679
≈ 2207
Therefore, the rate of change of weekly sales with respect to time when t = 0 is approximately 2207 units per month.
Hence, the correct option is C.
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