The rate at which water is sprayed on a field of vegetables is given by \(R(t)=3\sqrt{1+3t^3}\), where \(t\) is in minutes and \(R(t)\) is in gallons per minute. During the time interval \(0\leq t \leq 6\), what is the average rate of water flow, in gallons per minutes?

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The average value of the rate is given by:

$$ \frac{1}{b-a} \int\limits_a^b R(t)   dt $$ $$ = \frac{1}{6-0} \int\limits_0^6 3\sqrt{1+3t^3}   dt $$

Evaluating this with a calculator:

$$ = \frac{1}{6} \int\limits_0^6 3\sqrt{1+3t^3}   dt \approx \boxed{31.012} $$