In 2011 a​ country's federal receipts​ (money taken​ in) totaled ​$2.20
 
trillion. In​ 2013, total federal receipts were ​$2.81
 
trillion. Assume that the growth of federal​ receipts, F, can be modeled by an exponential function and use 2011 as the base year ​(tequals
​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 ​$12
 
​trillion?
Question content area bottom
Part 1
​a) Find the growth rate k.
kequals
enter your response here
​(Do not round until the final answer. Then round to six decimal places as​ needed.)
Part 2
Write the exponential function​ F(t).
​F(t)equals
enter your response here
Part 3
​b) Estimate total federal receipts in 2015.
The total federal receipts in 2015 will be approximately ​$enter your response here
 
trillion.
​(Use the answer from part​ (a) to find this answer. Round to two decimal places as​ needed.)
Part 4
​c) When will total federal receipts be ​$12
 
​trillion?
The total federal receipts will be ​$12
 
trillion approximately enter your response here
 
years after 2011.
​(Use the answer from part​ (a) to find this answer. Round to one decimal place as​ needed.)

This problem involves modeling the growth of federal receipts using an exponential growth model. Let's break it down into parts and solve each one step by step.

### Exponential Growth Model:
The general formula for exponential growth is:

\[
F(t) = F_0 \cdot e^{kt}
\]

Where:
- $F(t)$ is the amount of federal receipts at time $t$ (in trillions of dollars).
- $F_0$ is the initial amount of federal receipts at $t = 0$ (in 2011).
- $k$ is the growth rate.
- $t$ is the number of years after 2011.
- $e$ is Euler's number (approximately 2.71828).

We are given:
- In 2011 ($t = 0$), $F(0) = 2.20$ trillion dollars.
- In 2013 ($t = 2$), $F(2) = 2.81$ trillion dollars.

### Part 1: Find the Growth Rate $k$

We start with the exponential growth model and substitute the known values to find $k$.

\[
F(t) = F_0 \cdot e^{kt}
\]

Substitute $F(2) = 2.81$, $F_0 = 2.20$, and $t = 2$:

\[
2.81 = 2.20 \cdot e^{2k}
\]

Now solve for $k$:

1. Divide both sides by 2.20:
\[
\frac{2.81}{2.20} = e^{2k}
\]
\[
1.27727 = e^{2k}
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(1.27727) = 2k
\]
\[
0.245 = 2k
\]

3. Solve for $k$:
\[
k = \frac{0.245}{2} = 0.1225
\]

Thus, the growth rate $k$ to six decimal places is:
\[
k = 0.122499
\]

### Part 2: Write the Exponential Function

Now that we have the growth rate, the exponential function $F(t)$ becomes:

\[
F(t) = 2.20 \cdot e^{0.122499t}
\]

This is the exponential function that models the federal receipts in trillions of dollars.

### Part 3: Estimate Total Federal Receipts in 2015

To estimate the total federal receipts in 2015, we substitute $t = 4$ (since 2015 is 4 years after 2011) into the exponential function:

\[
F(4) = 2.20 \cdot e^{0.122499 \cdot 4}
\]

First, calculate the exponent:
\[
0.122499 \times 4 = 0.489996
\]

Now calculate $e^{0.489996}$:
\[
e^{0.489996} \approx 1.63272
\]

Finally, multiply by 2.20:
\[
F(4) = 2.20 \cdot 1.63272 \approx 3.59
\]

So, the estimated total federal receipts in 2015 will be approximately **$3.59 trillion**.

### Part 4: When Will Total Federal Receipts Be $12 Trillion?

To find when the total federal receipts will be $12 trillion, we set $F(t) = 12$ and solve for $t$:

\[
12 = 2.20 \cdot e^{0.122499t}
\]

1. Divide both sides by 2.20:
\[
\frac{12}{2.20} = e^{0.122499t}
\]
\[
5.4545 = e^{0.122499t}
\]

2. Take the natural logarithm of both sides:
\[
\ln(5.4545) = 0.122499t
\]
\[
1.696 = 0.122499t
\]

3. Solve for $t$:
\[
t = \frac{1.696}{0.122499} \approx 13.85
\]

So, the total federal receipts will reach $12 trillion approximately **13.9 years after 2011**, or in the year **2025**.

---

### Summary:
- **Growth rate $k$**: $0.122499$
- **Exponential function**: $F(t) = 2.20 \cdot e^{0.122499t}$
- **Estimated federal receipts in 2015**: $3.59 trillion
- **When federal receipts will be $12 trillion**: Around 13.9 years after 2011, or in the year 2025.

---

Do you need any clarifications or more details? Here are some related questions you might find useful:

1. How does exponential growth compare to linear growth?
2. What is the significance of the constant $e$ in exponential growth?
3. How would the receipts change if the growth rate $k$ were doubled?
4. How can logarithms be used to solve more complex growth problems?
5. What other real-world phenomena can be modeled by exponential growth?

**Tip**: When working with exponential functions, always pay attention to whether the variable is in the exponent or outside—it drastically changes how you approach the problem.

In 2011 a country's federal receipts (money taken in) totaled $2.20 trillion. In 2013, total federal receipts were $2.81 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 $12 trillion?

Solve an exponential growth problem involving federal receipts from 2011 to 2013, finding the growth rate, estimating future receipts for 2015, and determining when receipts will reach $12 trillion. Includes detailed step-by-step solutions using exponential growth models and logarithms. Suitable for high school students in Grades 10-12.

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