Questions: (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).
74.4 / (1+8.22 e^(-0.47 * 11))
The amount spent is approximately 71.1 billion in 2011 and 74.4 billion in 2022.
(b) Use the graph to determine the first full year in which expenditures for food and nutrition assistance were higher than 74 billion.
Assume the model remains accurate and graph the function from the year 2000 to the year 2030. Choose the correct graph.
A. 0 <= x <= 30 and 0 <= y <= 100 with X scl=3 and Yscl =10
B. 0 <= x <= 30 and 0 <= y <= 100 with X scl=3 and Yscl =10
C. 0 <= x <= 100 and 0 <= y <= 30 with X scl=10 and Yscl=3
D. 0 <= x <= 30 and 0 <= y <= 100 with X scl=3 and Yscl=10
Transcript text: (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).
\[
\frac{74.4}{1+8.22 e^{-0.47 \cdot 11}}
\]
The amount spent is approximately $\$ 71.1$ billion in 2011 and $\$ 74.4$ billion in 2022.
(b) Use the graph to determine the first full year in which expenditures for food and nutrition assistance were higher than $\$ 74$ billion.
Assume the model remains accurate and graph the function from the year 2000 to the year 2030. Choose the correct graph.
A.
$\square$
$0 \leq x \leq 30$ and $0 \leq y \leq 100$ with $X s c l=3$ and Yscl $=10$
B.
$0 \leq x \leq 30$ and $0 \leq y \leq 100$ with $X s c l=3$ and Yscl $=10$
c.
$0 \leq x \leq 100$ and $0 \leq y \leq 30$ with $X s c l=10$ and $\mathrm{Yscl}=3$
D.
$0 \leq x \leq 30$ and $0 \leq y \leq 100$ with $X s c l=3$ and $\mathrm{Yscl}=10$
Solution
Solution Steps
Step 1: Find the expression for the amount spent in 2011
Let $x$ be the number of years since 2000. The expression for the amount spent in billions of dollars is given by:
$\frac{74.4}{1 + 8.22e^{-0.47x}}$
For the year 2011, $x = 2011 - 2000 = 11$.
Substituting $x = 11$ into the given expression:
$\frac{74.4}{1 + 8.22e^{-0.47 \times 11}}$
The amount spent in 2011 is approximately $71.1 billion.
Step 3: Calculate the amount spent in 2022
For the year 2022, $x = 2022 - 2000 = 22$.
Substituting $x=22$ into the given expression:
$\frac{74.4}{1 + 8.22e^{-0.47 \times 22}} \approx \frac{74.4}{1 + 8.22e^{-10.34}} \approx \frac{74.4}{1 + 8.22 \times 0.00003} \approx \frac{74.4}{1+0.0002466} \approx 74.4$
The amount spent in 2022 is approximately $74.4 billion.
Step 4: Determine the correct graph and first year expenditures exceed $74 billion
We are looking for a graph that represents the given function from $x=0$ (year 2000) to $x=30$ (year 2030). The function is increasing as x increases, and it approaches 74.4. Option B correctly reflects this behavior.
To find the first year where the expenditures are greater than $74 billion, we can visually inspect the graph in option B. The graph crosses $y=74$ a little after $x=20$. Since $x$ represents years after 2000, $x=20$ corresponds to the year 2020. The first full year after that is 2021.
Final Answer:
The amount spent in 2011 is $71.1 billion, and the amount spent in 2022 is $74.4 billion. The correct graph is B. The first full year in which expenditures exceed \$74 billion is 2021.