For a certain island country, the current population is 2.2 million people and is expected to double every 35 years. Presently, the country's food supply can feed 3.8 million people and is projected to grow by an additional 25,000 people each year. Determine the year in which the country will begin to experience food shortages. Justify your answer. 


To determine the year when the country will begin to experience food shortages, we need to model both the population growth and the growth of the food supply over time.

### Step 1: Model Population Growth
The population doubles every 35 years. This type of growth is exponential and can be modeled using the formula:

\[
P(t) = P_0 \cdot 2^{\frac{t}{T}}
\]

Where:
- \( P(t) \) is the population at time \( t \) years from now.
- \( P_0 = 2.2 \) million is the current population.
- \( T = 35 \) years is the doubling time.
- \( t \) is the number of years from the present.

Thus, the population formula becomes:

\[
P(t) = 2.2 \cdot 2^{\frac{t}{35}}
\]

### Step 2: Model Food Supply Growth
The food supply grows linearly, starting from 3.8 million people and increasing by 25,000 people each year. The formula for the food supply \( F(t) \) at time \( t \) is:

\[
F(t) = 3.8 + 0.025 \cdot t
\]

Where:
- \( F(t) \) is the food supply (in millions of people) at time \( t \).
- 3.8 is the current food supply in millions.
- 0.025 is the yearly increase in the food supply (since 25,000 people = 0.025 million people per year).

### Step 3: Determine When Population Exceeds Food Supply
We want to find the year when the population exceeds the food supply, i.e., when:

\[
P(t) > F(t)
\]

Substitute the formulas for \( P(t) \) and \( F(t) \):

\[
2.2 \cdot 2^{\frac{t}{35}} > 3.8 + 0.025 \cdot t
\]

This inequality can be solved numerically to find the value of \( t \).

### Step 4: Numerical Solution
I'll solve this equation to find the value of \( t \).The population will exceed the food supply approximately 39.18 years from now. 

### Step 5: Determine the Year
Assuming the current year is \( t = 0 \), the food shortage will begin in about 39 years. If the present year is 2024, the country will experience food shortages around the year:

\[
2024 + 39.18 \approx 2063
\]

Thus, the country will begin to experience food shortages around the year **2063**.

Would you like a deeper explanation of the calculations or any further clarifications?

Here are five related questions you might find helpful:
1. What are some real-world examples of exponential growth in populations?
2. How does the concept of carrying capacity relate to food supply shortages?
3. What factors can slow down exponential population growth in a country?
4. Can linear growth of resources be sustainable long-term for an exponentially growing population?
5. How can governments or organizations plan for potential food shortages?

**Tip:** Exponential growth often appears deceptively slow at first, but it accelerates rapidly in later stages, so planning for shortages should start early.

For a certain island country, the current population is 2.2 million people and is expected to double every 35 years. Presently, the country's food supply can feed 3.8 million people and is projected to grow by an additional 25,000 people each year. Determine the year in which the country will begin to experience food shortages. Justify your answer.

Explore how to determine when a country's population will surpass its food supply using exponential population growth and linear food supply growth models. The problem involves a population doubling every 35 years and a food supply growing by 25,000 people annually. Discover when food shortages will begin in this detailed step-by-step solution, ideal for students in Grades 9-12.

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