Exponential/Logarithmic Word Problems

Complex numbers, rational functions, logarithms, sequences and series, matrix operations, etc.
mrwisdom
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Exponential/Logarithmic Word Problems

Postby mrwisdom » Wed May 04, 2011 10:04 pm

1. The township of brock is growing at a rate of 6.5% per annum. How many people are there in brock township now, if there will be 15000 in 4.5 years?

i have tried using A=Pert to solve this, but im a little lost do i convert the growth rate(r) to 0.065? should i be using a different forumla instead?



2. If the population of bacteria doubles every 30 min, how long would it take for the population to triple?

I decided to start with 100 bacteria and follow the A=Pert formula. Still not sure the format for r. But i did something like the lesson on exponential word problems with. just like thinking about it you would assume 45 mins but i don't know how to do it mathematically other then

300=100e0.5t
3=e0.5t
ln(3)/0.5=t
t=2.197

THis as my attempt at it and its probably very wrong.

Of course if the format for r i have used is wrong then the answer will be wrong. Also i need to rearrange the A=Perk formula for time after i have found the constant, and im stumped on what it should look like.

thanks

nona.m.nona
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Re: Exponential/Logarithmic Word Problems

Postby nona.m.nona » Thu May 05, 2011 11:47 am

mrwisdom wrote:1. The township of brock is growing at a rate of 6.5% per annum. How many people are there in brock township now, if there will be 15000 in 4.5 years?

Since this is "per annum" growth (that is, with time periods of one year) rather than "continuously", you may be expected to use the compound-interest formula, A = P(1 + r/n)^(nt). In this case, A = 15,000, r = 0.065, n = 1, and t = 4.5. Solve the equation for the value of t.

mrwisdom wrote:2. If the population of bacteria doubles every 30 min, how long would it take for the population to triple?

You can use A = Pe^(rt) in the manner explained here.

In particular, use the half-life information (in this case, the doubling-time information) to find the growth constant: P = 1 (100% of the original amount), A = 2 (200% of the original amount), and t = 0.5 (half an hour, or you could use "30" for "thirty minutes"). Then solve for the value of r.

Then plug back into the formula, using P = 1, A = 3, and the value you found for r. Solve for t.

mrwisdom
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Joined: Thu Oct 08, 2009 2:39 am
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Re: Exponential/Logarithmic Word Problems

Postby mrwisdom » Thu May 05, 2011 1:38 pm

mrwisdom wrote:1. The township of brock is growing at a rate of 6.5% per annum. How many people are there in brock township now, if there will be 15000 in 4.5 years?


I tried using the Compund-interest formula and could not get it to work with what i was doing, so i used the A=Pert
After some playing around with is i had it solve for P by A=Pert --> to P=A/ert which allowed me to get 11,195.92 for the starting population.

mrwisdom wrote:2. If the population of bacteria doubles every 30 min, how long would it take for the population to triple?


And i followed what you said for this question ending up with 47.54 minutes

Thank you. :)


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