Final answer:
To find the rate of change of profit with respect to time, differentiate the profit function and multiply it by the rate at which the number of units produced changes with respect to time. However, without additional information, we cannot determine the rate of change of profit with respect to time.
Step-by-step explanation:
To find the rate of change of the profit with respect to time, we need to calculate the derivative of the profit function. The profit function (P) is given by P(x) = p(x) - c(x), where p(x) is the revenue function and c(x) is the cost function.
The revenue function is equal to the price (p) times the number of units produced (x): p(x) = (1.700 - 0.016x)x. The cost function is c(x) = 715.000 + 240x.
To find dP/dt, we need to differentiate the profit function P(x). The derivative of P(x) is dP/dx = dp/dx - dc/dx.
Let's differentiate the revenue and cost functions:
dp/dx = (1.700 - 0.032x) dx/dx = 1.700 - 0.032x
dc/dx = 240
Now, substitute the derivatives into the derivative of the profit function:
dP/dx = dp/dx - dc/dx = (1.700 - 0.032x) - 240 = -0.032x - 238.3
The rate of change of the profit with respect to time, dP/dt, is equal to dP/dx times dx/dt. Since x represents the number of units produced in a week, dx/dt is the rate at which the number of units produced changes with respect to time. Without additional information, we cannot determine dx/dt, so we cannot find dP/dt.