Question
At the beginning of a population study, a city had 230,000 people. Each year since, the population has grown by \( 8.2 \% \). Let \( t \) be the number of years since start of the study. Let \( y \) be the city's population. Write an exponential function showing the relationship between \( y \) and \( t \).
Ask by Lindsey Tucker. in the United States
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The population \( y \) after \( t \) years is given by \( y = 230000 \times (1.082)^t \).
Solution
To write an exponential function that describes the relationship between the city's population \( y \) and the number of years \( t \) since the start of the study, we can use the formula for exponential growth:
\[
y = y_0 \cdot (1 + r)^t
\]
where:
- \( y_0 \) is the initial population,
- \( r \) is the growth rate (as a decimal),
- \( t \) is the time in years.
Given:
- The initial population \( y_0 = 230,000 \),
- The growth rate \( r = 8.2\% = 0.082 \).
Now, substituting these values into the formula:
\[
y = 230000 \cdot (1 + 0.082)^t
\]
This simplifies to:
\[
y = 230000 \cdot (1.082)^t
\]
Thus, the exponential function showing the relationship between \( y \) and \( t \) is:
\[
y = 230000 \cdot (1.082)^t
\]
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Simplify this solution Bonus Knowledge
The exponential function modeling the city's population over time can be expressed as \( y = 230,000(1 + 0.082)^t \) or simplified to \( y = 230,000(1.082)^t \). Here, \( t \) represents the number of years since the study began, and \( 1.082 \) reflects the annual growth factor due to the \( 8.2\% \) growth rate. To visualize this growth, imagine a small tree that seems to sprout only a little each year but eventually transforms into a towering giant. The same logic applies here: while the growth in the early years may seem small, compounded annually, it leads to significant changes down the line, showcasing the power of exponential growth!
