The population P (in thousands) of Reno, Nevada from 2000 through 2007 can be modeled by $P = 346.8 e ^ { k t }$ where t represents the year, with $t = 0$ corresponding to 2000.In 2005, the population of Reno was about 395,000.According to the model, during what year will the population reach 486,000.00?
A)2013
B)2005
C)2021
D)2017
E)2009

Question

The population P (in thousands) of Reno, Nevada from 2000 through 2007 can be modeled by $P = 346.8 e ^ { k t }$ where t represents the year, with $t = 0$ corresponding to 2000.In 2005, the population of Reno was about 395,000.According to the model, during what year will the population reach 486,000.00?  
A)2013
B)2005
C)2021
D)2017
E)2009

Quizplus · Accepted Answer

To find the year when the population will reach 486,000, we use the given model $P = 346.8e^{kt}$ and the information that in 2005 ($t = 5$), the population was 395,000. First, we solve for $k$ using the 2005 data:$$395 = 346.8e^{5k}$$Solving for $k$ gives us the growth rate. Then, we use the population of 486,000 to find $t$:$$486 = 346.8e^{kt}$$Solving this equation for $t$ will give us the year when the population reaches 486,000. The calculations show that this occurs in 2013.

The Population P (In Thousands) of Reno, Nevada from 2000 P=346.8ektP = 346.8 e ^ { k t }P=346.8ekt

The Population P (In Thousands) of Reno, Nevada from 2000 $P = 346.8 e ^ { k t }$