Suppose the world population in millions in the second half of the 20th century can be modeled by the equation 
P(t) = 2,560e^0.017185t.
 Use this equation to estimate the average world population to the nearest million during the time period of 1950 to 1990. (Note that 
t = 0 is 1950.)

To estimate the average world population between 1950 and 1990, we need to find the average value of the population function \( P(t) \) over the time interval from \( t = 0 \) (1950) to \( t = 40 \) (1990). The given population function is:

\[
P(t) = 2,560e^{0.017185t}
\]

The formula for the average value of a continuous function \( f(t) \) over the interval \([a, b]\) is:

\[
\text{Average Value} = \frac{1}{b - a} \int_{a}^{b} f(t) \, dt
\]

Here, \( f(t) = P(t) \), \( a = 0 \), and \( b = 40 \). Let's compute the average population by solving the following:

\[
\text{Average Population} = \frac{1}{40 - 0} \int_{0}^{40} 2,560e^{0.017185t} \, dt
\]

First, we compute the integral:

\[
\int 2,560e^{0.017185t} \, dt = \frac{2,560}{0.017185}e^{0.017185t} + C
\]

Now, applying the limits:

\[
\left[ \frac{2,560}{0.017185}e^{0.017185t} \right]_{0}^{40}
\]

Let me calculate this value.The estimated average world population during the time period from 1950 to 1990 is approximately **3,681 million** (rounded to the nearest million).

Would you like more details or have any questions about the calculation?

Here are 5 related questions you might find interesting:
1. How does the exponential growth model compare to other population growth models?
2. What would be the world population in 1970 using this model?
3. How can we adjust this model to account for factors like birth rates or migration?
4. How sensitive is the population estimate to changes in the growth rate parameter?
5. How would the average population change if we considered a different time interval, say from 1960 to 2000?

**Tip:** Exponential growth models are widely used in population dynamics, but they often overestimate populations in the long term due to environmental constraints.

Suppose the world population in millions in the second half of the 20th century can be modeled by the equation P(t) = 2,560e^0.017185t. Use this equation to estimate the average world population to the nearest million during the time period of 1950 to 1990. (Note that t = 0 is 1950.)

This page provides a detailed solution to estimating the average world population from 1950 to 1990 using the exponential growth model P(t) = 2,560e^0.017185t. Learn how to apply the average value formula and solve a definite integral. Suitable for high school students studying advanced mathematics.

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