A city had a population of 9,826 at the begining of 1950 and has been growing at 7.7% per year since then.

(a) Find the size of the city at the beginning of 1974.

To solve this problem, we use the formula for exponential population growth:

\[
P(t) = P_0 \times (1 + r)^t
\]

Where:
- \( P(t) \) is the population at time \( t \).
- \( P_0 \) is the initial population (in this case, 9,826 in 1950).
- \( r \) is the growth rate per year (7.7%, or 0.077).
- \( t \) is the number of years since the initial year (1974 - 1950 = 24 years).

### Step-by-step solution:
1. Identify the variables:
   - \( P_0 = 9826 \)
   - \( r = 0.077 \)
   - \( t = 1974 - 1950 = 24 \)

2. Plug these values into the population growth formula:
\[
P(24) = 9826 \times (1 + 0.077)^{24}
\]

3. Simplify the expression and calculate the population in 1974.

Let's compute this.The population of the city at the beginning of 1974 is approximately 58,285 people.

Would you like further details on the calculation or have any other questions? Here are some related questions to explore:

1. What would the population be at the beginning of 1980?
2. How long would it take for the population to double from 9,826?
3. What is the population growth in absolute terms from 1950 to 1974?
4. If the growth rate changed to 5% after 1974, what would the population be in 2000?
5. How does the population growth rate impact the city’s size over a longer period, say 100 years?

**Tip**: In exponential growth problems, even small changes in the rate can lead to significant differences over time.

A city had a population of 9,826 at the beginning of 1950 and has been growing at 7.7% per year since then. (a) Find the size of the city at the beginning of 1974.

Learn how to calculate the population growth of a city using exponential growth. This example starts with a population of 9,826 in 1950, growing at 7.7% per year, and explores the population size in 1974. This step-by-step solution involves the exponential growth formula and is suitable for students in Grades 9-12.

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