Part 1 of 3
Points: 0 of 1
A small company of science writers found that its rate of profit (in thousands of dollars) after $t$ years of operation is given by $P^{\prime}(t)=(7 t+14)\left(t^{2}+4 t+3\right)^{\frac{1}{6}}$.
(a) Find the total profit in the first four years.
(b) Find the profit in the seventh year of operation.
(c) What is happening to the annual profit over the long run?

Step 1 ：Given the rate of profit function \(P^{\prime}(t)=(7 t+14)\left(t^{2}+4 t+3\right)^{\frac{1}{6}}\)

Step 2 ：To find the total profit in the first four years, we need to integrate the rate of profit function from 0 to 4. The integral of a function gives the area under the curve, which in this case represents the total profit over a certain period of time.

Step 3 ：Let's denote the time as \(t\), and the rate of profit as \(profit\_rate = (7*t + 14)*(t**2 + 4*t + 3)**0.166666666666667\)

Step 4 ：By calculating the integral, we find that the total profit in the first four years is approximately 179.095506461639 thousand dollars.

Step 5 ：So, the final answer is: The total profit in the first four years is \(\boxed{179.095506461639}\) thousand dollars.