Question

Use an exponential model to solve the problem:
The population of Knoxville is 500,000 and is increasing at the rate of 3.25% each year. Approximately when will the population reach 1 million?

Answer 1

@satellite73 please help

Answer 2

the population starts at 500000, lets just say at t=0

after the first year, t=1
the population is increased by 3.25%
so the population would be 500000 + 500000 ( .0325)
or
500000(1.0325)

lets consider the next year t=2

the population increases another 3.25% from the last year
so the population will now be
500000(1.0325)(1.0325)
or
\[500000(1.0325)^2\]

Answer 3

now considering the 3rd year, t=3

it would be 500000(1.0325)(1.0325)(1.0325)

hopefully you see the pattern by now

so if we were to write this as an equation for "t" years later it would look like

\[P= 500000 (1.0325)^t\]
where P is the total population at "t" years later

Answer 4

in the case of your question,

you want to determine the number of years it takes for the initial population of 500000 to reach 1 mil

in which case, P= 1000000, and solve for t

Answer 5

any questions?

Answer 6

just one... for the equation
1000000=500000(1.0325)^t
how do I get the exponent down?

Answer 7

natural log rules
specifically power rule

\[\ln ( x ^y) = y \ln (x)\]

Answer 8

got it thanks