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Question 6: A local university has a current enrollment of 10,000 students. The enrollment is increasing at a rate of
1.5% each year. Find the number of years it will take for the population to increase to 12,000 students?

Question $6$ : A local university has a current enrollment of $10$ , $000$ students. The enrollment is increasing at a rate of $1.5 \%$ each year. Find the number of years it will take for the population to increase to $12$ , $000$ students?

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Exponential growth and decay: word problems

Full solution

Q. Question

6

: A local university has a current enrollment of

10

,

000

students. The enrollment is increasing at a rate of

1.5 \%

each year. Find the number of years it will take for the population to increase to

12

,

000

students?

Identify Data: Identify the initial population $P_0$ , final population $P$ , and growth rate $r$ . $P_0 = 10,000$ students, $P = 12,000$ students, $r = 1.5\%$ or $0.015$ as a decimal.
Use Exponential Growth Formula: Use the formula for exponential growth: $P = P_0 \times (1 + r)^t$ , where $t$ is the number of years. $\newline$ We need to solve for $t$ .
Rearrange Formula for $t$ : Rearrange the formula to solve for $t$ : $t = \frac{\log(\frac{P}{P_0})}{\log(1 + r)}$ .
Substitute Values: Substitute the known values into the rearranged formula: $t = \frac{\log(\frac{12,000}{10,000})}{\log(1 + 0.015)}$ .
Calculate t Values: Calculate the values: $t = \frac{\log(1.2)}{\log(1.015)}$ .
Use Calculator: Use a calculator to find the values: $t \approx \frac{\log(1.2)}{\log(1.015)} \approx \frac{0.07918}{0.00645}.$
Complete Calculation: Complete the calculation: $t \approx 12.27$ . Since we can't have a fraction of a year, we round up to the next whole number. $t = 13$ years.

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