Find the rate of change of total revenue, cost, and profit with respect to time. Assume that $R(x)$ and $C(x)$ are in dollars. $R(x)=3 x, \quad C(x)=0.01 x^{2}+0.5 x+20$, when $x=25$ and $\frac{d x}{d t}=8$ units per day The rate of change of total revenue is $\$$ per day.

Step 1 ：Given that the revenue function is \(R(x) = 3x\), the cost function is \(C(x) = 0.01x^2 + 0.5x + 20\), and \(\frac{dx}{dt} = 8\) units per day when \(x = 25\).

Step 2 ：First, we find the rate of change of total revenue with respect to time. This can be found by taking the derivative of the revenue function with respect to x, and then multiplying by the rate of change of x with respect to time. This is because of the chain rule in calculus, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, the outer function is the revenue function and the inner function is x with respect to time.

Step 3 ：Taking the derivative of the revenue function \(R(x) = 3x\) with respect to x, we get \(\frac{dR}{dx} = 3\).

Step 4 ：Multiplying this by the rate of change of x with respect to time, \(\frac{dx}{dt} = 8\), we get \(\frac{dR}{dt} = 3 \times 8 = 24\).

Step 5 ：Final Answer: The rate of change of total revenue is \(\boxed{24}\) dollars per day.