Question

a population of a certain city increases exponentially at a rate of 6.5%. initially, it is 1.50 million. after how many years will the population reach 2.31 million?

6.546

6.643

6.123

6.142

Explanation:

Step1: Recall exponential growth formula

The exponential growth formula is $P(t) = P_0(1 + r)^t$, where $P(t)$ is the population at time $t$, $P_0$ is the initial population, $r$ is the growth rate (in decimal), and $t$ is the time in years.
Here, $P_0 = 1.50$ million, $r = 0.065$ (since $6.5\% = 0.065$), and $P(t) = 2.31$ million. So we substitute these values into the formula:
$2.31 = 1.50(1 + 0.065)^t$

Step2: Simplify the equation

First, divide both sides by $1.50$:
$\frac{2.31}{1.50}=(1.065)^t$
Calculate $\frac{2.31}{1.50}=1.54$
So the equation becomes $1.54 = 1.065^t$

Step3: Take natural logarithm of both sides

Take $\ln$ of both sides: $\ln(1.54)=\ln(1.065^t)$
Using the logarithm power rule $\ln(a^b)=b\ln(a)$, we get:
$\ln(1.54)=t\ln(1.065)$

Step4: Solve for t

Divide both sides by $\ln(1.065)$:
$t=\frac{\ln(1.54)}{\ln(1.065)}$
Calculate $\ln(1.54)\approx0.4317$ and $\ln(1.065)\approx0.0630$
Then $t=\frac{0.4317}{0.0630}\approx6.852$ Wait, wait, maybe I miscalculated. Wait, let's recalculate $\ln(1.54)$ and $\ln(1.065)$ more accurately.
Using calculator: $\ln(1.54)\approx0.431749$, $\ln(1.065)\approx0.063003$
So $t=\frac{0.431749}{0.063003}\approx6.853$? Wait, but the options are around 6.1 - 6.6. Wait, maybe I made a mistake in the formula? Wait, maybe it's continuous growth? Wait, the problem says "increases exponentially"—maybe it's continuous? Wait, no, the formula $P(t)=P_0e^{rt}$ is for continuous, but the problem says "at a rate of 6.5%", maybe it's discrete? Wait, but let's check again. Wait, maybe I messed up the division. Wait, 2.31 divided by 1.50: 1.50*1.54=2.31, correct. Wait, maybe the growth rate is 6.5% but the formula is $P(t)=P_0(1 + r)^t$. Wait, let's recalculate the logs. Wait, maybe my calculator is wrong. Wait, let's use common logarithm (base 10) instead. $\log(1.54)\approx0.1875$, $\log(1.065)\approx0.0273$, then $t=\frac{0.1875}{0.0273}\approx6.868$. Hmm, still not matching. Wait, maybe the problem is with the initial numbers? Wait, the initial population is 1.50 million, final is 2.31 million. Let's check 1.50*(1.065)^6: 1.065^6≈1.065*1.065=1.1342, *1.065=1.2079, *1.065=1.2864, *1.065=1.3700, *1.065=1.4590, *1.065=1.5538. Then 1.50*1.5538≈2.3307, which is more than 2.31. So at t=6, 1.50*(1.065)^6≈2.33, which is close to 2.31. Wait, maybe my calculation of the log was wrong. Wait, let's use t=6.643: 1.065^6.643. Let's calculate 6.643*ln(1.065)=6.643*0.063003≈0.4185. Then e^0.4185≈1.519, 1.50*1.519≈2.278, no. Wait, maybe the formula is continuous? Let's try continuous growth: $P(t)=P_0e^{rt}$. Then 2.31=1.50e^{0.065t}. Divide by 1.50: 1.54=e^{0.065t}. Take ln: ln(1.54)=0.065t. t=ln(1.54)/0.065≈0.4317/0.065≈6.642. Ah! There we go. So maybe the problem is using continuous exponential growth (since "exponentially" can sometimes refer to continuous). So the formula should be $P(t)=P_0e^{rt}$. That makes sense. So let's redo the steps with continuous growth.

Step1 (corrected): Use continuous growth formula

The continuous exponential growth formula is $P(t) = P_0e^{rt}$, where $r$ is the growth rate (in decimal). So $P_0 = 1.50$, $r = 0.065$, $P(t)=2.31$. Substitute:
$2.31 = 1.50e^{0.065t}$

Step2 (corrected): Simplify

Divide both sides by 1.50:
$\frac{2.31}{1.50}=e^{0.065t}$
$\frac{2.31}{1.50}=1.54 = e^{0.065t}$

Step3 (corrected): Take natural log

Take $\ln$ of both sides:
$\ln(1.54)=\ln(e^{0.065t})$
Since $\ln(e^x)=x$, this simplifies to:
$\ln(1.54)=0.065t$

Step4 (corrected): Solve for t

Divide both sides by 0.065:
$t=\frac{\ln(1.54)}{0.065}$
Calculate $\ln(1.54)\approx0.4317$
Then $t=\frac{0.4317}{0.065}\approx6.642$

Ah, that matches the option 6.643 (due to rounding differences). So the mistake was initially using the discrete growth formula instead of continuous, but since the problem says "exponentially" which often implies continuous growth (or maybe the problem expects the continuous formula here). So that's the correct approach.

Answer:

6.643 (the option with 6.643)