In 2011 a country's federal receipts (money taken in) totaled $\$ 2.37$ trillion. In 2013 , total federal receipts were $\$ 2.96$ trillion. Assume that the growth of federal receipts, $F$, can be modeled by an 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 $F(t)$.
\[
F(\mathrm{t})=2.37 \cdot e^{.111150 \mathrm{t}}
\]
b) Estimate total federal receipts in 2015.
The total federal receipts in 2015 will be approximately $\$ \square$ trillion.
(Use the answer from part (a) to find this answer. Round to two decimal places as needed.)

Step 1 ：We are given that in 2011 (base year, t=0), the federal receipts were $2.37 trillion, and in 2013 (t=2), the receipts were $2.96 trillion. We are to assume that the growth of federal receipts, F, can be modeled by an exponential function.

Step 2 ：The general form of an exponential function is \(F(t) = F(0) \cdot e^{kt}\), where F(t) is the amount at time t, F(0) is the initial amount, k is the growth rate, and e is the base of the natural logarithm.

Step 3 ：We can substitute the given values into this formula: \(2.96 = 2.37 \cdot e^{2k}\). Solving this equation for k, we get \(k = 0.111150\) to six decimal places.

Step 4 ：Substituting this value of k into the exponential function, we get \(F(t) = 2.37 \cdot e^{0.111150t}\).

Step 5 ：To estimate the total federal receipts in 2015, we substitute t = 4 (since 2015 is 4 years after the base year 2011) into the exponential function and calculate the value. This gives us \(F(4) = 2.37 \cdot e^{0.111150 \cdot 4} = 3.70\) trillion dollars.