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To solve this problem, we need to construct a quadratic model for the given population data and use it to predict the population in 2050.

The points given in the problem are:
- \( (0, 25) \): 1900, 25 million people.
- \( (50, 200) \): 1950, 200 million people.
- \( (100, 325) \): 2000, 325 million people.

Let:
- \( x \) be the number of years since 1900.
- \( y \) be the population in millions.

We want to find a quadratic equation of the form:
\[
y = ax^2 + bx + c
\]
that passes through the points \( (0, 25) \), \( (50, 200) \), and \( (100, 325) \).

### Step 1: Set Up the System of Equations
For each point, substitute \( x \) and \( y \) into the quadratic equation.

1. **Point (0, 25):**
   \[
   25 = a(0)^2 + b(0) + c \Rightarrow c = 25
   \]

2. **Point (50, 200):**
   \[
   200 = a(50)^2 + b(50) + 25
   \]
   Simplifying, we get:
   \[
   200 = 2500a + 50b + 25
   \]
   \[
   175 = 2500a + 50b
   \]

3. **Point (100, 325):**
   \[
   325 = a(100)^2 + b(100) + 25
   \]
   Simplifying, we get:
   \[
   325 = 10000a + 100b + 25
   \]
   \[
   300 = 10000a + 100b
   \]

### Step 2: Solve the System of Equations
Now we have two equations:

1. \( 175 = 2500a + 50b \)
2. \( 300 = 10000a + 100b \)

We can simplify both equations:
- Divide the first equation by 25:
  \[
  7 = 100a + 2b
  \]
- Divide the second equation by 100:
  \[
  3 = 100a + b
  \]

Now we have a simpler system:

1. \( 7 = 100a + 2b \)
2. \( 3 = 100a + b \)

Subtract the second equation from the first:
\[
4 = b
\]

Now substitute \( b = 4 \) into the second equation:
\[
3 = 100a + 4
\]
\[
100a = -1
\]
\[
a = -0.01
\]

### Step 3: Write the Quadratic Equation
Now we have:
\[
a = -0.01, \quad b = 4, \quad c = 25
\]
Thus, the quadratic model is:
\[
y = -0.01x^2 + 4x + 25
\]

### Step 4: Estimate the Population in 2050
To find the population in 2050, we substitute \( x = 150 \) (since 2050 is 150 years after 1900):
\[
y = -0.01(150)^2 + 4(150) + 25
\]
\[
y = -0.01(22500) + 600 + 25
\]
\[
y = -225 + 600 + 25
\]
\[
y = 400
\]

### Answer
The estimated population in 2050 is **400 million**.

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Would you like further details or have any questions about this solution?

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Here are five related questions that expand on this concept:

1. How would the model change if a different type of function (e.g., exponential) was used?
2. What assumptions are made when using quadratic models to predict population growth?
3. How does this model handle data points that don’t fit perfectly on a quadratic curve?
4. Can this model predict accurately far beyond the year 2050? Why or why not?
5. How would we adjust the model if a new data point, say for 2010, was added?

**Tip:** Always double-check each calculation step-by-step, especially when solving systems of equations, to ensure accuracy in building models.

The population of a certain country was approximately 25 million in 1900, 200 million in 1950, and 325 million in 2000. Construct a model for this data by finding a quadratic equation whose graph passes through the points (0, 25), (50, 200), and (100, 325). Use this model to estimate the population in 2050.

Learn how to construct a quadratic model for population growth data based on given points. This problem involves creating a quadratic equation and using it to estimate future population. Suitable for high school students studying quadratic modeling and systems of equations.

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