Final answer:
To determine the rate of change of profit with respect to time, the derivative of the profit function P(x) is found and evaluated at x=42. This value is multiplied by the given dt/dx, resulting in a rate of change of 720 dollars per unit of time when x=42 and dt/dx=90.
Step-by-step explanation:
The student is asking about the rate of change of total profit with respect to time when given the revenue function R(x) and cost function C(x), for a specific value of x and a value of dt/dx. To find this, we need to determine the profit function P(x), which is defined as P(x) = R(x) - C(x). We then find P'(x), the derivative of the profit function with respect to x, to know the rate of change of profit with respect to quantity (x), and finally, use the given value of dt/dx to find the rate of change of profit with respect to time when x = 42.
Using the given functions: R(x) = 90x - 0.5x2 and C(x) = 40x + 7, we first find P(x):
P(x) = R(x) - C(x) = (90x - 0.5x2) - (40x + 7).
Then we find P'(x) by differentiating P(x):
P'(x) = dP/dx = d(R(x) - C(x))/dx = d(90x - 0.5x2 - 40x - 7)/dx = 50 - x.
So when x = 42, P'(42) = 50 - 42 = 8. This is the rate of change of profit with respect to quantity at x = 42. To find the rate of change of profit with respect to time, we use P'(x) and multiply it by dt/dx, which gives us P'(x) * (dt/dx):
P'(42) * dt/dx = 8 * 90 = 720.
Therefore, when x = 42 and dt/dx = 90, the rate of change of total profit with respect to time is 720 dollars per unit of time.