Anthropologists determine that new dwellings in an ancient farming community were constructed monthly as modeled by the function f(x)=2x+100, where x is the current month of the year and f(x) is the number of dwellings constructed by the end of that month. Additionally, they determine that the population grew exponentially each month, thanks to the discovery of more fertile land for farming. This growth is modeled by the equation `g(x)=3^x`, where g(x) represents the current population at the end of a given month. What is the smallest integer value of x, with 1 representing the end of January and 12 representing the end of December, at which the population surpasses the number of dwellings built?

Answer

To solve the problem, we need to find the smallest integer value of \( x \) (representing the month) such that the population \( g(x) \) surpasses the number of dwellings constructed \( f(x) \). The functions are given as:

- \( f(x) = 2x + 100 \) (number of dwellings)
- \( g(x) = 3^x \) (population)

We want to find the smallest \( x \) such that:

\[ g(x) > f(x) \]

This translates to:

\[ 3^x > 2x + 100 \]

We will evaluate this inequality for integer values of \( x \) from 1 to 12 (representing January to December).

### Step-by-step Solution:

1. **Evaluate \( x = 1 \)**:
\[ g(1) = 3^1 = 3 \]
\[ f(1) = 2(1) + 100 = 102 \]
Since \( 3 < 102 \), \( x \) cannot be 1.

2. **Evaluate \( x = 2 \)**:
\[ g(2) = 3^2 = 9 \]
\[ f(2) = 2(2) + 100 = 104 \]
Since \( 9 < 104 \), \( x \) cannot be 2.

3. **Evaluate \( x = 3 \)**:
\[ g(3) = 3^3 = 27 \]
\[ f(3) = 2(3) + 100 = 106 \]
Since \( 27 < 106 \), \( x \) cannot be 3.

4. **Evaluate \( x = 4 \)**:
\[ g(4) = 3^4 = 81 \]
\[ f(4) = 2(4) + 100 = 108 \]
Since \( 81 < 108 \), \( x \) cannot be 4.

5. **Evaluate \( x = 5 \)**:
\[ g(5) = 3^5 = 243 \]
\[ f(5) = 2(5) + 100 = 110 \]
Since \( 243 > 110 \), \( x = 5 \) is a valid solution.

6. **Check \( x = 6 \)** (for completeness):
\[ g(6) = 3^6 = 729 \]
\[ f(6) = 2(6) + 100 = 112 \]
Since \( 729 > 112 \), \( x = 6 \) is also valid, but we are looking for the smallest \( x \).

Thus, the smallest integer value of \( x \) for which the population surpasses the number of dwellings is:

\[ \boxed{5} \]