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Answer :
To find the current rate of change of sales, we need to differentiate the sales function with respect to time. In this case, the rate of change of sales with respect to time can be calculated as the derivative of the sales function with respect to x, multiplied by the rate of change of x with respect to time.
Given:
s = 60000 - 390000 e^(-0.0007x) (sales function)
x = 2000 + 300t (advertising costs)
We will first differentiate the sales function with respect to x:
ds/dx = d/dx (60000 - 390000 e^(-0.0007x))
= 0 - 390000 (-0.0007) e^(-0.0007x)
= 273 e^(-0.0007x)
Next, we will differentiate x with respect to time:
dx/dt = d/dt (2000 + 300t)
= 300
Finally, we can calculate the current rate of change of sales by evaluating ds/dt at the current values:
ds/dt = (ds/dx) * (dx/dt)
= 273 e^(-0.0007x) * 300
Substituting x = 2000 into the equation, we get:
ds/dt = 273 e^(-0.0007 * 2000) * 300
Calculating this expression will give you the current rate of change of sales.
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