Step 1 :Given the growth rate \(k = 0.11115\) and the initial total federal receipts \(F_0 = 2.37\) trillion dollars in 2011.
Step 2 :We can write the exponential function for total receipts as \(F(t) = F_0 \cdot e^{kt} = 2.37 \cdot e^{0.11115t}\).
Step 3 :To estimate the total federal receipts in 2015, we substitute \(t = 4\) (since 2015 is 4 years after the base year 2011) into the function \(F(t)\).
Step 4 :Calculating this gives us an estimated total federal receipts in 2015 of approximately \$3.70 trillion.
Step 5 :To find out when the total federal receipts will be \$11 trillion, we set \(F(t) = 11\) and solve for \(t\).
Step 6 :Solving this gives us \(t \approx 13.81\). Since our base year is 2011, we add this to 2011 to find the year.
Step 7 :This gives us the year as approximately 2025.
Step 8 :So, the final answers are: The estimated total federal receipts in 2015 will be approximately \(\boxed{3.70}\) trillion dollars. The total federal receipts will be \$11 trillion approximately in the year \(\boxed{2025}\).