Assume the annual rate of change in the national debt of a country (in billions of dollars per year) can be modeled by the function \[ D^{\prime}(t)=850.29+822.46 t-190.62 t^{2}+16.4 t^{3} \] where $t$ is the number of years since 1995. By how much did the debt increase between 1996 and 2006 ? The debt increased by $\$$ billion. (Round to two decimal places as needed.)

Step 1 ：Assume the annual rate of change in the national debt of a country (in billions of dollars per year) can be modeled by the function \(D^{\prime}(t)=850.29+822.46 t-190.62 t^{2}+16.4 t^{3}\) where \(t\) is the number of years since 1995.

Step 2 ：We need to find the increase in debt between 1996 and 2006. To do this, we find the integral of the rate of change function from 1 to 11 (since 1996 is 1 year after 1995 and 2006 is 11 years after 1995). The integral of a rate of change function gives the total change over the interval of integration.

Step 3 ：The result from the calculation is the total increase in the national debt from 1996 to 2006.

Step 4 ：Final Answer: The debt increased by \(\boxed{33366.30}\) billion dollars.