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# Exponential growth & decay word problems

## Video transcript

let's do a couple of word problems dealing with exponential growth and decay so this first problem suppose a radioactive substance decays at a rate of 3.5% per hour what percent of the substance is left after six hours so let's make a little table here to just imagine just imagine what's going on and then we'll try to come up with a formula for in general how much is left after n hours so let's say hours that have passed by and percentage left four cent left so after zero hours what percent is left well has it decayed yet so we have 100% left after one hour what's happened once it decays at a rate of 3.5% per hour so 3.5% is gone or another way to think about it is 0.965 right if you remember if you take 1 minus 3.5% or if you take 100% minus 3.5% this is how much we're losing every hour that equals 96.5% so each hour we're going to have 96.5% of the previous hour so an hour one we're going to have ninety six point five percent of our zero or zero point nine six five times 100 times our zero now what happens in our to our two well we're going to have ninety six point five percent of the previous hour we will have lost three-and-a-half percent which means that we have 96 and a half percent of the previous hour so it'll be zero point nine six five times this times zero point nine six five times 100 I think you see where this is going in general so in the first hour we have 0.965 to the first power times 100 in the zeroth hour we have 0.965 to the zeroth power we don't see it but there's a 1 there times 100 and this in the second hour 0.965 to the second power times 100 so in general in the nth hour let me do this in a nice bold color in the nth hour we're going to have 0.965 to the nth power times 100 left of our of our radioactive substance and often times you'll see it written this way you have your initial amount times your common ratio 0.965 to the nth power this is how much we're going to have left after n hours well now we can answer the question after 6 hours how much are we going to have left well we're going to have 100 times 0.965 to the sixth power left and we could use a calculator to figure out what that is let's use our trusty calculator so we have 100 times 0.965 to the sixth power which is equal to this is equal to 80 point seven five this is all in percentages so it's 80 point seven five percent of our original substance so eighty point seven five percent let's do another one of these so we have Nadia owns a chain of fast-food restaurants that operated 200 stores in 1999 in 1999 if the rate of increase is actually this a typo here should be 8% the rate of increase is 8% annually how many stores does the restaurant operate in 2007 so let's think about the same thing in 1999 so let's say years years after 1999 and let's talk about how many stores how many stores Nadia is operating her fast-food chain so 1999 1999 itself is zero years after 1999 and she is operating 200 stores then in 2000 which is one year after 1999 how many is she going to be operating well she grows at the rate of 8% annually so she'll be operating all the stores that she had before all the stores that she had before plus 8% of the stores she had before so 1.08 times the number of stores she had before now you're gonna see the common ratio here's 1.08 if you're growing by 8 percent that's equivalent to multiplying by 1.08 let me make that clear 200 plus 0.08 times 200 well this is just 1 times 1 times 200 plus 0.08 times 200 that's 1.08 times 200 then in 2001 in 2001 what's going on this is now two years after 1999 is you're going to grow 8 percent from this number so you're going to multiply 1.0 8 times that number times 1.08 times 200 i think you get the general gist if after n years after 1999 it's going to be 1.08 let me write it this way it's going to be 200 times 1.08 to the nth power after 2 years one point eight squared one year one point eight to the first power zero years this is the same thing as a 1 times 200 which is 1.08 to the 0th power so they're asking us how many stores does a restaurant operate in 2007 well 2007 this is 2007 is eight years after 1999 so here n is equal to 8 so let us substitute n is equal to 8 the answer to our question will be 200 times 1.08 to the 8th power let's get our calculator out and calculate it so we want to figure out 200 200 times 1.0 eight to the eighth power to the eighth power she is going to be operating 370 restaurants and she'll be in process of opening a few more so she's going to be if we round it down she's going to be operating 370 restaurants so 8% growth might not look like something that that's that's so fast or that exciting but in under a decade in only eight years she would have gotten her restaurant chain from 200 to 370 restaurants so for over eight years you see that the compounding growth by 8% actually ends up being quite dramatic