Question

The population of a particular town is given by the function P(t) = 1,200(1.05)2t, where t is the time in years and P(t) is the population after t years. What is the current population, the percentage growth rate, and the population size (rounded to the nearest whole person) in 7 years?

Answer 1

Hint: Percentage Growth Rate = r ⋅ 100 in A = A0(1 + r)t

1,200, 5% annually, and 1,689
1,200, 50% semiannually, and 20,503
1,200, 10.5% per year, and 2,414
1,200, 5% semiannually, and 2,376

Answer 2

Hey random cooool person.
So we have our function,\[\large\rm P(t)=1200(1.05)^{2t}\]

Answer 3

They gave us a really helpful hint,
When the function is in this form:\[\large\rm P(t)=P_o(1+r)^{t}\]Then \(\large\rm 100r\) is our growth rate.

Notice, however, that we have a 2 in the exponent of our problem.
That's not the form that we want.
So we'll have to apply a handy exponent rule to fix that.\[\large\rm (a^b)^c=a^{bc}\]

Answer 4

So we can use this rule to rewrite our function as,\[\large\rm P(t)=1200\left[(1.05)^2\right]^{t}\]Squaring the 1.05 gives us,\[\large\rm P(t)=1200\left(1.1025\right)^t\]

Answer 5

Oh they gave us semi-annually for some of the options :(
That's really annoying... ugh

Answer 6

back my internet died on me.

Answer 7

I guess we actually don't need to do all this extra work since we have multiple choice.\[\large\rm P(\color{orangered}{t})=1200(1.05)^{2\color{orangered}{t}}\]We can simply plug in t=7 and simplify,\[\large\rm P(\color{orangered}{7})=1200(1.05)^{2\color{orangered}{\cdot7}}\]You'll need a calculator for this :)

Answer 8

My bad my parents friends just left i had to help escort them, alright so continue :`> whats next? after i calculate it?

Answer 9

Calculate it :U

Answer 10

It should match up with one of your options :)

Answer 11

:o oh alright thanks. got it.