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CCSS.Math:

there are a hundred and seventy deer on our reservation the deer population is increasing at a rate of 30 percent per year write a function that gives the deer population P of T on the reservation T years from now all right let's think about it I don't like always pause this video and see if you can work it out on your own but let's think about what P is zero is P is zero this is going to be the initial population of deer the population at time zero well we know that that's going to be the hundred seventy deer that we start on the reservation now let's think about what P of one is what's going to be the population after one year well it's going to be our original population 170 but that it increases at a rate of thirty percent per year so it's going to be one seventy plus another 30 percent of one seventy so I could write that as 30 percent times one seventy or I could write this as one seventy plus 0.3 times one seventy thirty percent as a decimal is the same thing as 30 hundredths or three tenths or I could write this as if i factor out a 170 I would get one seventy times one plus 0.3 which is the same thing as one seventy times one point zero three and this is a really good thing to take a hard look at because you'll see it a lot when we're growing by a certain rate when we're dealing with what turns out to be exponential functions when we if we are growing so I almost made a mistake there it's one point three almost so here you go one point three one plus zero point three is one point three so once again take a hard look at this right over here because it's going to be something that you see a lot with exponential functions when you grow by 30% that means you keep your hundred percent that you had before and then you add another 30 percent and so you would multiply your original quantity by one hundred and thirty percent and one hundred thirty percent is the same thing as point three so if you are growing by 30% you are growing by 3/10 you would multiply your initial quantity by 1.3 so let's use that idea to keep going so what is the population after 2 years well you would start that second year with the population at the end of 1 year so it's going to be that 170 times 1.3 and then over that year you're going to grow by another 30% so if you're going to grow by another 30% that's equivalent to multiplying by 1.3 again or you could say that this is equal to 170 times 1 point 3 to the second power and so I think you see where this is going if we wanted to write a general P of T so if we just want to write a general P of T it's going to be whatever we started with 170 and we're going to multiply that by 1 point 3 however many times however many years have gone by so to the T power because for every year we grow by 30% which is equivalent mathematically to multiplying by 1.3 so after a hundred years it would be 170 times 1.3 to the hundredth power