The population of a mid-sized city is growing exponentially. The population of the city was 250,000 in the year 2000 and 273,400 in 2004. Find the exponential growth function that models the population growth of the city. Use to represent the year 2000.

Question

The population of a mid-sized city is growing exponentially. The population of the city was 250,000 in the year 2000 and 273,400 in 2004. Find the exponential growth function that models the population growth of the city. Use  to represent the year 2000.

Quizplus · Accepted Answer

Let's use the formula for exponential growth:
P(t) = P0 * e^(rt)

where:
P(t) is the population at time t
P0 is the initial population (in 2000)
r is the annual growth rate (we don't know this yet)
t is the time in years (measured from 2000)

We know that P0 = 250,000 and P(4) = 273,400 (since the population in 2004 was 273,400, which is 4 years after 2000). Using these values, we can solve for r:

P(t) = 250,000 * e^(rt)
P(4) = 273,400 = 250,000 * e^(4r)
e^(4r) = 273,400/250,000
4r = ln(273,400/250,000)
r = ln(273,400/250,000) / 4
r = 0.016

Now we know that the annual growth rate is 0.016. Plugging this into the formula, we get:

P(t) = 250,000 * e^(0.016t)

Finally, using 11ebfdc1_ec36_77a5_abe3_5d95a8e63006_TBX8727_11 to represent the year 2000 (or t=0), we get:

P(t) = 250,000 * e^(0.016(t-11ebfdc1_ec36_77a5_abe3_5d95a8e63006_TBX8727_11))

which simplifies to:

P(t) = 250,000 * e^(0.016t-31.463)

Therefore, the exponential growth function that models the population growth of the city is:

A) 11ebfdc1_ec36_77a6_abe3_ad6d2572eab6_TBX8727_11

The Population of a Mid-Sized City Is Growing Exponentially