Based on the U.S. censuses taken from 1850 through 2000, the population of the United States can be approximated by the equation
y = 0.00701x2 − 25.311x + 22850.323
where x is the year and y is the approximate U.S. population in millions. Using this equation, find the approximate population for the following years. (Round your answers to one decimal place.)
(a)    1880
 million


To find the approximate population of the United States in the year 1880 using the given quadratic equation, we'll substitute \( x = 1880 \) into the equation:

\[
y = 0.00701x^2 - 25.311x + 22850.323
\]

Substituting \( x = 1880 \):

\[
y = 0.00701 \times (1880)^2 - 25.311 \times 1880 + 22850.323
\]

Let's calculate this step by step:

1. Calculate \( 1880^2 \):
   \[
   1880^2 = 3534400
   \]

2. Multiply \( 0.00701 \times 3534400 \):
   \[
   0.00701 \times 3534400 = 24788.144
   \]

3. Calculate \( 25.311 \times 1880 \):
   \[
   25.311 \times 1880 = 47584.68
   \]

4. Plug these values back into the equation for \( y \):
   \[
   y = 24788.144 - 47584.68 + 22850.323
   \]

5. Simplify the expression:
   \[
   y = 24788.144 - 47584.68 + 22850.323 = 6,053.787
   \]

Rounding to one decimal place:

\[
y \approx 6053.8 \, \text{million}
\]

So, the approximate population of the United States in the year 1880 was **6053.8 million** (6.05 billion). 

Would you like to solve for other years, or do you have any questions?

### Related Questions:
1. What would be the approximate population for the year 1900 using the same equation?
2. How does the equation approximate the population trend over time? 
3. What is the population growth rate implied by this equation between 1850 and 2000?
4. If the population in 2000 was known to be around 281 million, how accurate is the approximation?
5. Can this model be used to predict future populations accurately? Why or why not?

**Tip:** When working with quadratic equations for predictions, always consider the domain and the range to understand the accuracy and relevance of your results.

Learn how to approximate the U.S. population in 1880 using a quadratic equation derived from U.S. census data from 1850 to 2000. The equation y = 0.00701x^2 − 25.311x + 22850.323 is used, providing an approximation rounded to one decimal place. Suitable for advanced high school students and beyond.

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