Let's go through each part of the problem step by step:

Given Information:

• Population of San Francisco in 2010: $805,235$
• Population of San Francisco in 2020: $873,965$

Let $t$ be the number of years after 2010. So, $t = 0$ corresponds to the year 2010 and $t = 10$ corresponds to the year 2020.

Part a) Find a linear equation of the form $p(t) = mt + b$:

We need to find the slope $m$ and the intercept $b$ for the linear equation $p(t) = mt + b$, where $p(t)$ represents the population at time $t$.

1. Slope $m$: $m = \frac{\Delta p}{\Delta t} = \frac{873,965 - 805,235}{10 - 0} = \frac{68,730}{10} = 6,873$ The slope $m = 6,873$, which represents the population increase per year.

2. Intercept $b$: Since $b$ represents the population in 2010 (i.e., when $t = 0$): $b = 805,235$

Thus, the equation is: $p(t) = 6,873t + 805,235$

Part b) Estimate the population in the year 2026:

To estimate the population in 2026, we set $t = 2026 - 2010 = 16$: $p(16) = 6,873(16) + 805,235 = 109,968 + 805,235 = 915,203$

The estimated population in 2026 is 915,203.

Part c) Predict the year in which the population will reach 1,044,970:

We need to solve for $t$ when $p(t) = 1,044,970$: $1,044,970 = 6,873t + 805,235$ Subtract 805,235 from both sides: $1,044,970 - 805,235 = 6,873t$ $239,735 = 6,873t$ Solve for $t$: $t = \frac{239,735}{6,873} \approx 34.86$

Since $t \approx 34.86$, the year will be approximately $2010 + 35 = 2045$.

The population is expected to reach 1,044,970 in the year 2045.

Would you like further details or have any questions?

Here are 5 related questions to further explore this topic:

1. How would the equation change if the population growth rate increased after 2020?
2. What happens to the linear model if the population decreases instead of increasing?
3. How would we model the population if it followed exponential rather than linear growth?
4. How does rounding affect the accuracy of population predictions?
5. Can we calculate the population change over a different time interval, say from 2015 to 2025?

Tip: Linear models assume a constant rate of change, so they might not accurately capture population growth trends over very long periods.