Part 4 of 4
Points: 0 of 1 exponential function and use 2011 as the base year $(t=0)$.
a) Find the growth rate $k$ to six decimal places, and write the exponential function $F(t)$, for total receipts in trillions of dollars.
b) Estimate total federal receipts in 2015.
c) When will total federal receipts be $\$ 11$ trillion?
a) Find the growth rate $k$.
\[
k=.111150
\]
(Do not round until the final answer. Then round to six decimal places as needed.)
Write the exponential function $F(t)$.
\[
F(t)=2.37 \cdot e^{111150 t}
\]
b) Estimate total federal receipts in 2015.
The total federal receipts in 2015 will be approximately $\$ 3.70$ trillion.
(Use the answer from part (a) to find this answer. Round to two decimal places as needed.)
c) When will total federal receipts be $\$ 11$ trillion?
The total federal receipts will be $\$ 11$ trillion approximately years after 2011.
(Use the answer from part (a) to find this answer. Round to one decimal place as needed.)

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 \(\boxed{2025}\).