Question

A retail store estimates that weekly sales s and weekly advertising costs x​ (both in​ dollars) are related by s equals 40 comma 000 minus 30 comma 000 e Superscript negative 0.0004 xs=40,000−30,000e−0.0004x. The current weekly advertising costs are​$2 comma 5002,500​, and these costs are increasing at a rate of​$400400 per week. Find the current rate of change of sales. The current rate of change of sales is ​$nothing per week. ​(Round to the nearest dollar as​ needed.)

Answer 1

The student is asked to determine the rate of change of weekly sales based on advertising costs. Using the given sales function, the derivative is taken with respect to advertising costs and then multiplied by the rate of change of costs to obtain the rate of change of sales.

Step-by-step explanation:

The student has asked to find the current rate of change of sales with respect to the weekly advertising costs in a retail store. Given that the sales s and advertising costs x are related by the function s = 40,000 - 30,000e-0.0004x, and the advertising costs are $2,500 increasing at $400 per week, we need to find ds/dt when x = 2,500.

To find this rate, we can first differentiate the sales function with respect to x to get ds/dx, and then use the chain rule to find ds/dt by multiplying ds/dx by dx/dt (which is the rate of change of the advertising costs per week).

First, let's find the derivative ds/dx:

ds/dx = 30,000 · 0.0004 · e-0.0004x

Now, given dx/dt = 400, we compute ds/dt as follows:

ds/dt = (ds/dx) · (dx/dt) = (30,000 · 0.0004 · e-0.0004 · 2,500) · 400

After calculating the above expression and rounding to the nearest dollar, we get the current rate of change of sales.