The population of a city was 166 thousand at the begining of 2004. The exponential growth rate was 1.5% per year. Use the formula P(t)=P[o]e^(kt) where P[o] is the population in 2004 and k is the exponential growth rate.

a) predict the population in 2016, to the nearest thousand.
b) during which year will the population reach 258 thousand?

I will assume that t is the time in years since 2004

P(t) = 166 e^(.015t)

a)
so for 2016 , t = 12

P(12) = 166 e^(12(.015)) = 166 e^.18 = 198.7 thousand or 199 thousand

b) 258 = 166 e^.015t
1.55422 = e^.015t
.015t = ln 1.55422
t = ln1.55422/.015 = 29.4 years since 2004
or in the year 2033

To solve this problem, we can use the formula for exponential growth:

P(t) = P[o] * e^(kt)

Where:
P(t) is the population at time t
P[o] is the population at the initial time (2004 in this case)
k is the exponential growth rate
t is the time in years

a) To predict the population in 2016, we need to find P(2016), given that P[o] = 166 thousand and k = 1.5%.

First, we need to convert the growth rate from a percentage to a decimal. So, 1.5% = 0.015.

Plugging in the values, we have:
P(2016) = 166 * e^(0.015 * 2016)

Using a calculator or software program, we can evaluate this expression to find the result. Rounding to the nearest thousand, we get:
P(2016) ≈ 166 * e^(30.24) ≈ 281 thousand

Therefore, the predicted population in 2016 is approximately 281 thousand.

b) To determine during which year the population will reach 258 thousand, we need to solve the equation P(t) = 258.

258 = 166 * e^(0.015t)

Dividing both sides of the equation by 166, we have:
e^(0.015t) ≈ 258/166

To isolate the exponential term, we take the natural logarithm (ln) of both sides:

ln(e^(0.015t)) ≈ ln(258/166)

Simplifying, we get:
0.015t ≈ ln(258/166)

Finally, we solve for t by dividing both sides by 0.015:
t ≈ ln(258/166) / 0.015

Evaluating this expression using a calculator or software program, we find:
t ≈ 16.904

Therefore, the population will reach 258 thousand approximately during the 17th year, or 2021.