In 2011 a country's federal receipts (money taken in) totaled $\mathrm{\$}2.37$ trillion. In 2013 , total federal receipts were $\mathrm{\$}2.96$ 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 $\mathrm{\$}11$ 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 $\mathrm{\$}◻$ 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 $2.37trillion,andin2013(t=2),thereceiptswere$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.