A city population, which was initially 15,500, has been dropping by 3% a year. Write an exponential function and graph the function. Use your graph to predict when the population will drop below 8,000.

Could someone help me solve this?

Question

A city population, which was initially 15,500, has been dropping by 3% a year. Write an exponential function and graph the function. Use your graph to predict when the population will drop below 8,000. 

Could someone help me solve this?

whpalmer4 · Answer

If it is dropping by 3% each year, that means after 1 year, it has 97% of the previous year's population, or 0.97*previous year's population.  If we take $P_0$ to be the initial population, after 1 year, we have $P(1) = P_0*0.97$
After two years, we have $P(2) = P(1)*0.97 = P_0*0.97*0.97 = P_0(0.97)^2$
After three years, we have $P(3) = P(2)*0.97 = P_0(0.97)^2*0.97 = P_0(0.97)^3$

Do you see a pattern developing here?

anonymous · Answer

yess

whpalmer4 · Answer

Okay, so what is your formula for $$P(t)=$$

anonymous · Answer

P(t) = 15,500(0.97)t 
Right??

anonymous · Answer

So then,  the population will drop below 8,000 in 21.7 years right?

whpalmer4 · Answer

That appears to be correct!

anonymous · Answer

Awesome! That helps a lot! Thank you

anonymous · Answer

:)

the_fizicx99 · Answer

I thought about graphing it myself in my head being a simple function eh, but I forgot what the decay rate would be like >_< doing it algebraically would also give me the same answer but graphing it is faster.

the_fizicx99 · Answer

Nvm rolf just answered myself ._.

anonymous · Answer

hahaha :) silly