In 1990, a total of $578 billion was spent on food and drinks in a particular country. In 2003, the total spent was $913 billion.
(a)
Find the equation of the exponential function that can be used to model the total T spent (in billions of dollars) on food and drinks in this country as a function of the number of years t since 1990. (Round your decimal value to four decimal places.)
T = answer a
 
(b)
Use your model to predict the amount spent (in billions of dollars) in 2000. (Round your answer to the nearest integer.)
answer b
 billion dollars
(c)
What is your prediction for the total sales of food and drink (in billions of dollars) in 2022? (Round your answer to the nearest integer.)
answer c
 billion dollars
(d)
Estimate when the total sales will reach $2 trillion if this exponential trend continues. (Round your answer to two decimal places.)
t = answer d


Let's solve each part of this problem step by step.

### Part (a): Finding the Exponential Model

1. **Identify known values:**
   - In 1990 (t = 0), $ T = 578 $ billion dollars.
   - In 2003 (t = 13), $ T = 913 $ billion dollars.

2. **Model form:**
   The exponential growth model is:
   \[
   T = T_0 \cdot e^{kt}
   \]
   where $ T_0 $ is the initial amount (578 in 1990) and $ k $ is the growth rate we need to find.

3. **Set up equations:**
   - For 1990: $ T_0 = 578 $
   - For 2003: $ 913 = 578 \cdot e^{13k} $

4. **Solve for $ k $:**
   \[
   e^{13k} = \frac{913}{578}
   \]
   \[
   e^{13k} \approx 1.5796
   \]
   Taking the natural logarithm of both sides:
   \[
   13k = \ln(1.5796)
   \]
   \[
   k \approx \frac{\ln(1.5796)}{13} \approx 0.0353
   \]

5. **Exponential function:**
   \[
   T = 578 \cdot e^{0.0353t}
   \]

Thus, the equation for the model is:
\[
\boxed{T = 578 \cdot e^{0.0353t}}
\]

### Part (b): Predict the amount spent in 2000

1. **Determine $ t $:**
   Since 2000 is 10 years after 1990, $ t = 10 $.

2. **Substitute $ t = 10 $ into the model:**
   \[
   T = 578 \cdot e^{0.0353 \cdot 10}
   \]
   Calculating this:
   \[
   T \approx 578 \cdot e^{0.353} \approx 578 \cdot 1.4233 \approx 823
   \]

So, the predicted amount spent in 2000 is:
\[
\boxed{823} \text{ billion dollars}
\]

### Part (c): Prediction for the total spent in 2022

1. **Determine $ t $:**
   Since 2022 is 32 years after 1990, $ t = 32 $.

2. **Substitute $ t = 32 $ into the model:**
   \[
   T = 578 \cdot e^{0.0353 \cdot 32}
   \]
   Calculating this:
   \[
   T \approx 578 \cdot e^{1.1296} \approx 578 \cdot 3.0947 \approx 1789
   \]

So, the predicted total spent in 2022 is:
\[
\boxed{1789} \text{ billion dollars}
\]

### Part (d): Estimate when total sales will reach $2 trillion

1. **Set up the equation:**
   We want $ T = 2000 $ (since $2 trillion = 2000 billion).
   \[
   2000 = 578 \cdot e^{0.0353t}
   \]

2. **Solve for $ t $:**
   \[
   e^{0.0353t} = \frac{2000}{578}
   \]
   \[
   e^{0.0353t} \approx 3.4594
   \]
   Taking the natural logarithm of both sides:
   \[
   0.0353t = \ln(3.4594)
   \]
   \[
   t \approx \frac{\ln(3.4594)}{0.0353} \approx 35.50
   \]

So, the total sales will reach $2 trillion approximately:
\[
\boxed{35.50} \text{ years after 1990, or around 2025.}
\]

Let me know if you need more details, or if you have any questions. Here are five related questions to deepen your understanding:

1. How would the model change if we used a linear function instead of an exponential one?
2. What would the prediction be for other years, like 2015 or 2030, using this model?
3. How does the value of $ k $ affect the speed of growth in an exponential model?
4. Could this model apply to other industries, or would we need to adjust it for different sectors?
5. How accurate might this model be if the growth rate changes over time?

**Tip:** In exponential models, a small change in the growth rate $ k $ can lead to significant differences over time due to the compounding effect.

In 1990, a total of $578 billion was spent on food and drinks in a particular country. In 2003, the total spent was $913 billion.
(a) Find the equation of the exponential function that can be used to model the total T spent (in billions of dollars) on food and drinks in this country as a function of the number of years t since 1990. (Round your decimal value to four decimal places.)
T = answer a
(b) Use your model to predict the amount spent (in billions of dollars) in 2000. (Round your answer to the nearest integer.)
answer b billion dollars
(c) What is your prediction for the total sales of food and drink (in billions of dollars) in 2022? (Round your answer to the nearest integer.)
answer c billion dollars
(d) Estimate when the total sales will reach $2 trillion if this exponential trend continues. (Round your answer to two decimal places.)
t = answer d

Explore an exponential growth model based on food and drink spending trends from 1990 to 2003. This problem guides you through finding an exponential function to predict future spending, estimating sales for 2000 and 2022, and determining when spending will reach $2 trillion. Suitable for high school students studying exponential models.

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