4 Homework Part 2 of 4 Points: 0 of 1 Save In 2011 a country's federal receipts (money taken in) totaled $\$ 2.43$ trillion. In 2013, total federal receipts were $\$ 2.74$ trillion. Assume that the growth of federal receipts, $\mathrm{F}$, can be modeled by an exponential function and use 2011 as the base year $(\mathrm{t}=0)$. a) Find the growth rate $\mathrm{k}$ to six decimal places, and write the exponential function $\mathrm{F}(\mathrm{t})$, for total receipts in trillions of dollars. b) Estimate total federal receipts in 2015. c) When will total federal receipts be $\$ 9$ trillion? a) Find the growth rate $k$. \[ k=0.060033 \] (Do not round until the final answer. Then round to six decimal places as needed.) Write the exponential function $\mathrm{F}(\mathrm{t})$. \[ F(t)=\square \] ew an example Get more help - Clear all

Step 1 ：Given that the initial amount F(0) = 2.43 trillion dollars in 2011, and F(2) = 2.74 trillion dollars in 2013, we can use the formula for exponential growth F(t) = F(0) * e^(kt) to find the growth rate k. Here, F(t) is the amount at time t, F(0) is the initial amount, e is the base of the natural logarithm (approximately 2.71828), k is the growth rate, and t is the time.

Step 2 ：First, we plug these values into the formula and solve for k: \(2.74 = 2.43 * e^(2k)\)

Step 3 ：Next, we divide both sides by 2.43: \(1.12757 = e^(2k)\)

Step 4 ：Then, we take the natural logarithm of both sides: \(\ln(1.12757) = 2k\)

Step 5 ：Finally, we solve for k: \(k = \frac{\ln(1.12757)}{2} \approx 0.060033\)

Step 6 ：\(\boxed{k \approx 0.060033}\) is the growth rate.

Step 7 ：To estimate total federal receipts in 2015, we plug t = 4 (since 2015 is 4 years after the base year 2011) into the exponential function: \(F(4) = 2.43 * e^(0.060033 * 4) \approx 3.06\) trillion dollars

Step 8 ：\(\boxed{F(4) \approx 3.06}\) trillion dollars is the estimated total federal receipts in 2015.

Step 9 ：To find when total federal receipts will be 9 trillion dollars, we set F(t) = 9 and solve for t: \(9 = 2.43 * e^(0.060033 * t)\)

Step 10 ：Next, we divide both sides by 2.43: \(3.70576 = e^(0.060033 * t)\)

Step 11 ：Then, we take the natural logarithm of both sides: \(\ln(3.70576) = 0.060033 * t\)

Step 12 ：Finally, we solve for t: \(t = \frac{\ln(3.70576)}{0.060033} \approx 22.5\) years

Step 13 ：\(\boxed{t \approx 22.5}\) years after the base year 2011, which is around the year 2034, total federal receipts will be 9 trillion dollars.