A country has a food deficit of 10%. Its population grows continuously at a rate of 3% per year. Its

Q: Wait, why do we use the continuous growth model for population but the discrete compound model for food production?

This is a fantastic observation! In large-scale demographic modeling, population changes happen every second—people are born and pass away constantly—so we model it as a continuous process using $e^{rt}$. However, food production is typically measured in annual harvest cycles, which are discrete events. The problem explicitly defines the nature of these growths, and we must respect the mathematical model assigned to each.

Q: Why can we just cancel out $a$ and $P_0$? Is it possible they are zero?

That is a great logical check. In the context of a country, the population $P_0$ and the food requirement per person $a$ must be positive values (you cannot have a negative population or negative food needs). Because they are strictly positive, we are mathematically justified in dividing both sides of the inequality by $a P_0$ without flipping the inequality sign.

Q: Why is the initial food production $0.9 a P_0$? Shouldn't it be $0.1 a P_0$?

I understand the confusion! The problem states there is a $10\%$ deficit. A deficit means we are *missing* $10\%$ of what we need. If we need $100\%$ (or $1.0$) and we are missing $10\%$ ($0.1$), we are currently producing $1.0 - 0.1 = 0.9$ of the required amount. Always think in terms of 'what we have' versus 'what we need'.

Question

A country has a food deficit of 10%. Its population grows continuously at a rate of 3% per year. Its annual food production every year is 4% more than that of the last year. Assuming that the average food requirement per person remains constant, prove that the country will become self-sufficient in food after n years, where n is the smallest integer bigger than or equal to $\frac{\ln 10 - \ln 9}{\ln(1.04) - 0.03}$.

Sigma Percentile · Accepted Answer

### The Race Against Time: Modeling Sustainability

Imagine you are a policy advisor for a nation facing a critical challenge. You have two curves on your desk: one representing the relentless, continuous growth of your population, and the other representing the annual, discrete growth of your agricultural output.

The goal is to find the exact moment when your nation crosses the threshold into self-sufficiency. This is not just an algebra problem; it is a fundamental exercise in understanding the dynamics of growth.

### Phase 1: Defining the Players

Let us define our variables clearly. We start at time $t=0$ with an initial population $P_0$. Every citizen requires a constant amount of food, which we call $a$.

Therefore, the initial total food requirement is $R(0) = a P_0$. However, the country is currently in a deficit, producing $10\%$ less than it needs.

Mathematically, this means the initial production $F(0)$ is:
$$F(0) = a P_0 - 0.1 a P_0 = 0.9 a P_0$$
This $0.9$ is our starting point—our baseline efficiency.

### Phase 2: The Mathematical Arena

Now, the population grows continuously at a rate of $3\%$ per year. When you hear "continuously," your mind should immediately jump to the exponential growth formula:
$$P(t) = P_0 e^{0.03t}$$

Consequently, the total food requirement at any time $t$ is:
$$R(t) = a P_0 e^{0.03t}$$

On the other side of the equation, food production grows annually by $4\%$. This is a discrete compound growth model. The production at time $t$ is given by $F(t) = F(0)(1 + 0.04)^t$.

Substituting our initial production, we get:
$$F(t) = 0.9 a P_0 (1.04)^t$$

### Phase 3: The Inequality of Survival

To achieve self-sufficiency, we need our production to meet or exceed our requirements. We set up the inequality:
$$F(t) \geq R(t)$$

Substituting our expressions, we get:
$$0.9 a P_0 (1.04)^t \geq a P_0 e^{0.03t}$$

Here is the beauty of the problem: the initial conditions $a$ and $P_0$ are common to both sides. Since they are positive, we can divide them out. We are left with:
$$0.9 (1.04)^t \geq e^{0.03t}$$

This reveals a profound truth: the time it takes to become self-sufficient is independent of the initial population size or the specific food requirement per person! It depends entirely on the growth rates and the initial deficit.

### Phase 4: The Logarithmic Bridge

We have $t$ trapped in the exponents. To free it, we apply the natural logarithm ($\ln$) to both sides:
$$\ln(0.9 \cdot 1.04^t) \geq \ln(e^{0.03t})$$

Using the laws of logarithms, specifically $\ln(AB) = \ln A + \ln B$ and $\ln(x^y) = y \ln x$, the left side becomes $\ln(0.9) + t \ln(1.04)$. On the right side, the $\ln$ and $e$ cancel out, leaving us with $0.03t$.

Now, we group the $t$ terms:
$$t \ln(1.04) - 0.03t \geq -\ln(0.9)$$

Factoring out $t$, we get:
$$t [\ln(1.04) - 0.03] \geq -\ln(0.9)$$

### The Final Result

Recall that $-\ln(0.9) = \ln(0.9^{-1}) = \ln(\frac{1}{0.9}) = \ln(\frac{10}{9}) = \ln 10 - \ln 9$.

Thus, we arrive at the final expression for the time threshold:
$$t \geq \frac{\ln 10 - \ln 9}{\ln(1.04) - 0.03}$$

Since $t$ must be the smallest value satisfying this, we have proven our condition. You have successfully modeled the path to **sustainability**.

Visualized Solution

Initial Food Requirement

Initial Food Deficit

Continuous Population Growth

Discrete Food Production Growth

Condition for Self-Sufficiency

Equating Production and Requirement

Canceling Common Terms

Taking Natural Logarithm

Using Logarithmic Properties

Grouping $t$ Terms

Simplifying $- ln (0.9)$

Isolating $t$

The Sigma Insight: Linear Differential Equations

The Race Against Time

Phase 1

Phase 2

Phase 3

Phase 4

The Final Result

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