The amount of money a certain government spent on food and nutrition assistance​ (in billions of​ dollars) can be approximated by the function f left parenthesis x right parenthesis equals StartFraction 74.5 Over 1 plus 8.24 e Superscript negative 0.34 x EndFractionf(x)=
74.5
1+8.24e−0.34x​, where xequals=0 corresponds to the year 20002000. Use a graphing calculator or other technology to complete parts​ (a) and​ (b).
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Part 1
​(a) Find the amount spent in 20112011 and 20222022. 
Write the expression that can be used to find the amount spent in the year 20112011 ​(in billions of​ dollars).

The function provided to approximate the government spending on food and nutrition assistance is:

\[
f(x) = \frac{74.5}{1 + 8.24 e^{-0.34x}}
\]

Here, $ x = 0 $ corresponds to the year 2000, meaning that $ x = 11 $ corresponds to the year 2011, and $ x = 22 $ corresponds to the year 2022.

### To find the amount spent in 2011:

For the year 2011, $ x = 11 $, so the expression to calculate the spending in 2011 would be:

\[
f(11) = \frac{74.5}{1 + 8.24 e^{-0.34 \times 11}}
\]

### To find the amount spent in 2022:

For the year 2022, $ x = 22 $, so the expression to calculate the spending in 2022 would be:

\[
f(22) = \frac{74.5}{1 + 8.24 e^{-0.34 \times 22}}
\]

### Now let me compute the amounts for both 2011 and 2022.The calculated government spending on food and nutrition assistance is approximately:

- **In 2011:** \$62.30 billion
- **In 2022:** \$74.16 billion

Do you have any questions, or would you like more details on how these values were computed?

Here are 5 related questions you could explore:
1. How would the spending change if the exponential growth rate were different?
2. What year corresponds to when the spending reaches \$70 billion?
3. How would the graph of this function behave over the long term?
4. Can we estimate when the spending might level off or stabilize based on the function?
5. What does the parameter 74.5 represent in the context of this model?

**Tip:** Exponential models like this one often represent growth that slows down over time, approaching a maximum value.

The amount of money a certain government spent on food and nutrition assistance (in billions of dollars) can be approximated by the function f(x) = 74.5 / (1 + 8.24e^(-0.34x)), where x = 0 corresponds to the year 2000. Use a graphing calculator or other technology to complete parts (a) and (b). (a) Find the amount spent in 2011 and 2022. Write the expression that can be used to find the amount spent in the year 2011 (in billions of dollars).

Explore how to use an exponential decay function to model government spending on food and nutrition assistance between 2000 and 2022. Learn how to calculate spending in 2011 and 2022 using the given formula, with clear step-by-step solutions. Suitable for Grades 10-12.

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