Question

A retail sporting goods store estimates that weekly sales s and weekly advertising costs x are related by the equation s = 2250 + 50x + 0.35x2. the current weekly advertising costs are $1500, and these costs are increasing at a rate of $125 per week. find the current rate of change of the weekly sales with respect to time. please show step by step working.

Answer 1

To find the current rate of change of the weekly sales with respect to time, we need to find the derivative of the equation s = 2250 + 50x + 0.35x^2 with respect to x and then evaluate it at x = 1500.

The first step is to take the derivative of the equation:

s = 2250 + 50x + 0.35x^2

d(s)/dx = 50 + 0.7x

Next, we need to evaluate the derivative at x = 1500:

d(s)/dx = 50 + 0.7 * 1500 = 50 + 1050 = 1100

So, the current rate of change of the weekly sales with respect to time is 1100 dollars per week. This means that for every $125 increase in the weekly advertising costs, the weekly sales will increase by $1100.