If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Constructing exponential models: percent change

CCSS.Math:

## Video transcript

sheppy is an ecologist who studies the change in the narwhal population of the arctic ocean over time she observed that the population loses 5.6 percent of its size every 2.8 months the population of Nar whales can be modeled by a function n which depends on the amount of time T in months when chappy began the study she observed that there were eighty nine thousand narwhals in our whales in the Arctic Ocean right a function that models the population of Nar whales T months since the beginning of chappies daddy like always pause the video and see if you can do it on your own before we work through it together so now let's work through it together and to get my sense of what this function needs to do it's always valuable to to see to create a table for some interesting inputs for the function and seeing and seeing how the function should behave so first of all if T is in months and n of T is in n is the Met models n is the number of narwhals there are no heart whales so what what when T is equal to zero what is n of zero well we know at T equals zero there are 89 thousand NAR whales in the ocean so eighty-nine thousand and that was somewhat what's another interesting one well T is in months and we know that we the population decrease is 5.6 percent every two point eight months so let's think about when T is two point eight two point eight months well then the population should have gone down five point six percent so going down five point six percent is the same thing as retaining what what's what's one minus five point six percent retaining 94.4% let me be clear 100% if you lose five point six percent you are going to be left with 94.4% the point six plus 0.4 gets you to 95 plus another five years to 100 so this another way of saying this sentence that the population loses five point six percent of its size every two point eight months is to say that the population is ninety four percent ninety four point four percent of its size every two point eight months or shrinks 94.4% of its original size every or let me let me phrase this clearly after every two point eight months the population you could either say it shrinks five point six percent or you could say it has it's gone from its ninety four point four percent of the population at the beginning of those two point eight months so after two point eight months the population should be eighty-nine thousand times i could write x 94.4% or i could write x zero point nine four four now if we go if we go another two point eight months so two times two point eight i obviously could just write that is i could write that as five point six months but let me just write this as two point eight months where are we going to be we're going to be at eighty nine thousand times zero point nine four four this is where we were before at the beginning of this period and we're going to be ninety four point four percent of that so we're going to multiply by ninety four point four percent again or zero point nine four four again or we can just say x zero point nine four four squared and after three of these periods well we're going to be x zero point nine four four again so it's going to be eighty nine thousand times zero point nine four four squared times zero point nine four four which is going to be zero point nine four four to the third power and I think you might see what's going on here we have an exponential function at between every two point eight months we are multiplying by this common ratio of zero point nine four four and so we can write our function n of T our initial value is eighty nine thousand eighty nine thousand times zero point nine four four to the power of however many of these two point eight month periods we've gone so far so if we take the number of months and we divide by two point eight that's how many two point eight month periods we have got gone and so notice when T equals zero all of this turns into one you raise something to the zero power just becomes one you have eighty nine thousand when T is equal to two point eight this exponent is one and we're going to multiply by zero point nine four or once when T is 5.6 the exponent is going to be 2 and we're going to multiply by zero point nine four four twice and I'm just doing the values that make the exponent integers but it's going to work for the ones in between I encourage you to graph it or to try those values on a calculator if you like but there you have it we're done we have modeled our NAR wails so let me just underline that we're done