A retail store estimates that weekly sales s and weekly advertising costs x (both in dollars) are related by
s=50000−430000e^(−00009x)
The current weekly advertising costs are 2000 dollars and these costs are increasing at the rate of 300 dollars per week. Find the current rate of change of sales.
The current rate of change of sales is -12,600 dollars per week.
To find the current rate of change of sales, we need to differentiate the sales equation with respect to time.
Given: s = 50000 - 430000e^(-0.00009x)
Let's assume t represents time in weeks and x represents the advertising costs in dollars.
Given: dx/dt = 300 (advertising costs are increasing at the rate of 300 dollars per week)
To find the rate of change of sales, we need to differentiate s with respect to t using the Chain Rule.
ds/dt = ds/dx * dx/dt
To differentiate s with respect to x, we need to consider the function s = f(x) = 50000 - 430000e^(-0.00009x).
Differentiating s with respect to x, we have:
df/dx = d/dx(50000) - d/dx(430000e^(-0.00009x))
df/dx = 0 - (-0.00009 * 430000 * e^(-0.00009x))
df/dx = 0.00009 * 430000 * e^(-0.00009x)
Now, substituting dx/dt = 300 and df/dx into the equation ds/dt = ds/dx * dx/dt, we get:
ds/dt = (0.00009 * 430000 * e^(-0.00009x)) * 300
ds/dt = 1290 * e^(-0.00009x)
Therefore, the current rate of change of sales is 1290 * e^(-0.00009x) dollars per week.
To find the current rate of change of sales, we need to calculate the derivative of the sales function in respect to time.
Given that the weekly advertising costs are increasing at a rate of 300 dollars per week, we can express the current weekly advertising costs as a function of time, t:
x(t) = 2000 + 300t
Now, let's differentiate the sales function with respect to time:
ds/dt = d(50000 − 430000e^(-0.00009x))/dt
To find the derivative, we need to use the chain rule. Let's denote u = -0.00009x:
ds/du = d(50000 - 430000e^u)/du
Next, let's find the derivative of the function inside the parentheses with respect to u:
ds/du = d(50000)/du - d(430000e^u)/du
ds/du = 0 - d(430000e^u)/du
ds/du = -430000 * d(e^u)/du
Now, we need to differentiate e^u with respect to u. The derivative of e^u is just e^u:
ds/du = -430000 * e^u
Substituting back for u = -0.00009x, we have:
ds/du = -430000 * e^(-0.00009x)
Finally, we can substitute x(t) = 2000 + 300t into the expression to find the rate of change of sales, ds/dt:
ds/dt = -430000 * e^(-0.00009 * (2000 + 300t))
Now we have an expression for the rate of change of sales with respect to time, ds/dt, which we can use to calculate the current rate of change of sales by plugging in the current value of t.