Main content

## College Algebra

### Course: College Algebra > Unit 13

Lesson 5: Interpreting the rate of change of exponential models# Interpreting change in exponential models

CCSS.Math: , ,

Sal finds the factor by which a quantity changes over a single time unit in various exponential models.

## Want to join the conversation?

- Why 1.75 is as the same as 75% instead of 0.75?(24 votes)
- 75% does = 0.75

The problem is dealing with exponential growth in the number of branches.

So tree has its original number of branches (100% or 1) + growth of 75% (0.75)

1 (original) + 0.75 (growth) = new percentage = 1.75%

For example: 200 branches * 100% + 75% growth = 200(1) + 200 (0.75) = 200 (1.75) = 350

If the problem just used 0.75, then the number of branches would actually be shrinking by 25%

For example: 200 * 0.75 = 150 branches

This is fewer branches than we started with. So, the tree is not growing, its shrinking.

Hope this helps.(57 votes)

- 4:02Sal says the bear population shrinks by a factor of 2/3, but this statement is just a double negative, if something shrinks by a factor <1, then it increases. So would it shrink by the reciprocal of 2/3? (So 3/2), or would it shrink by the difference of 1 and the original factor added to 1. Which would be (1 - 2/3 + 1 or 4/3).(6 votes)
- No, shrinking by a factor of 2/3 means the original amount is MULTIPLIED by 2/3, which means you have 2/3 of the original left.(5 votes)

- I don't understand why the bear population would shrink by a factor of 2/3 rather than 1/3. When it asked, "what factor is the bear population shrinking by" isn't it asking for the factor in function of the population lost rather than the one remained?(6 votes)
- The question is asking for the factor. In this case the factor is (2/3). The population
*is*decreasing by 1/3, but to get the third, you would multiply by two thirds.

example: 1 * 2/3 = 2/3

The 1 has decreased by one third to a solution of 2/3(3 votes)

- In the practice "Interpret change in exponential models" when given a question about finding percentage, if we are handed sat (0.75), why would this not be 75% for an answer. I've tried looking at the explanation but it doesn't make sense.(4 votes)
- Why is Math so darn hard all the time?

:((3 votes) - How do you do the (0.81)^t ones, he doesn't explain that(3 votes)
- If you've got something like f(x) = 43*(0.81)^t, then it's like the second last example with 2/3 as a factor. If you have a factor that is smaller than 1, then the number still decreases.(1 vote)

- In the first exercise, something is increasing, so we get "...increases by a factor 1.5" or so. In the second exercise something decreases. Decreasing is the opposit of increasing, so we'd expect "...decreases nu a factor 1.5". Instead we get a decrease by a factor less than 1. Isn't a decrease by a factor less than one an increase? Just like an increase by a factor less than one is a decrease?(1 vote)
- Yes... to decrease a value using multiplication, you need to multiply by a value less than 1. Any value over 1 would increase the number.(4 votes)

- So if n is 1 or more it grows and if it is less than one it shrinks?(1 vote)
- Basically, if you have a function f that is exponential and can be expressed as:
`f(x) = a * b^x`

, b determines the factor.`If b < 1, then f(x) will decrease as x increases.`

If b > 1, then f(x) will increase as x increases.

If b = 1, then no matter what x is, 1^x = 1, so it stays flat.(4 votes)

- Does the population of bears shrink by a factor or 2/3 or 1/3? It seems if there are 2/3 as many bears the next year, the population shrank by 1/3.(1 vote)
- It did shrink by 1/3, but not by a factor of 1/3. A 'factor' is a term of multiplication. Since there are 2/3 as many bears, the population was multiplied by 2/3, so we say it shrunk by a factor of 2/3.(3 votes)

- How can exponential growth and decay help us solve problems in the real world? How does this math subject affect and impact us throughout our lives? What is an example of exponential growth or decay in the real world?(2 votes)
- As we see in the video, exponential growth and decay can help us measure anything from the amount of a drug left in the body to the amount of radiation in an area.(1 vote)

## Video transcript

- [Voiceover] So I've
taken some screenshots of the Khan Academy exercise
"Interpreting Rate of Change "for Exponential Models
in Terms of Change" Maybe they're going to change the title. It seems a little bit too long. But anyway, let's actually
just tackle these together. So the first of spring an entire field of flowering trees blossoms. The population of locusts
consuming these flowers rapidly increases as the trees blossom. The relationship between
the elapsed time t in days since the beginning of spring and the total number of locusts L of t, so the number of locusts
is going to be a function of the number of days that have elapsed since the beginning of spring, is modeled by the following function. So locusts as a function
of time is going to be 750 times 1.85 to the "t"th power. Complete the following sentence about the daily rate of change
of the locust population. Every day the locust population, well, every day think about
what's going to happen. I'll draw a little table, just to make it hopefully a little bit clearer. So, we draw a little bit of a table. So we'll put t and L of t. So when t is zero, so when
zero days have elapsed, well that's going to be
1.85 to the 0th power, that's just going to be one. So you're going to have 750
locusts right from the get go. Then when t equals one,
what's going to happen? Well then this is going to be 750 times 1.85 to the 1st power. So it's going to be times 1.85. When t is equal to two, what's L of t? It's going to be 750 times 1.85 squared. Well that's the same thing as 1.85 times 1.85. So notice. And this is just comes out of this being an exponential function. Every day you have 1.85 times as many as you had the day before. 1.85 We essentially take what
we had the day before and we multiply it by 1.85. And since 1.85 is larger than one, that's going to grow the
number of locusts we have. So this is going to grow. I'm actually not using, I'm
not on the website right now so that's why, normally there
would be a dropdown here. So I'm going to grow by a factor of, well I'm going to grow by
a factor of 1.85 every day. Let's do another one of these. All right. So this one tells us
that Vera is an ecologist who studies the rate of change in the bear population
of Siberia over time. The relationship between
the elapsed time t in years since Vera began studying the population and the total number of bears N of t is modeled by the following function. All right, fair enough. We've got an exponential thing going on. Complete the following sentence about the yearly rate of change
of the bear population. Let's just thing about it. Every year that passes, t as in years now, every years that passes is going to be two thirds times the year before. I could do that same table that I just did just to make that clear,
so let me do that. Whoops. Let me, let me make this clear. So, table. So this is t and this is N of t. When t is zero, N of t you're
going to have 2187 bears. So that's the first year that she began studying that population. Zero years since Vera began
studying the population. The first year is going to be 2187 times two thirds to the first
power, so times two thirds. The second year is going to be 2187 times two thirds to the second power. So that's just two
thirds times two thirds. So each successive year
you're going to have two thirds the bear
population of the year before. You're multiplying the
year before by two thirds. So every year the bear population shrinks, shrinks by a factor of, by a factor of two thirds. All right, let's do one more of these. So they tell us that
Akiba started studying how the number of branches
on his tree change over time. All right. The relationship between
the elapsed time t in years since Akiba started studying his tree and the total number
of its branches N of t is modeled by the following function. Compete the following sentence about the yearly percent change
in the number of branches. Every year, blank percent of branches are added or subtracted from
the total number of branches. Well I'll draw another table, although you might get used to just being able to look at this and say well look, each year you're going to have 1.75 times the number branches
you had the year before. And so we have 1.75 times the number of branches the year before, you have grown by 75%, and I'll make that a little bit clearer. So 75%, every year 75% percent of branches are added to the total number of branches. And I'll just draw that table again like I've done in the last two examples to make that hopefully clear. Okay, so this is t and this is N of t. So t equals zero, you have 42 branches. T equals one, it's going
to be 42 times 1.75. Times 1.75. When t equals two, it's going
to be 42 times 1.75 squared. 42 times 1.75 times 1.75. So every year you are multiplying times 1.75 so times 1.75, something funky is happening
with my pen right over there. But if you're multiplying by 1.75, if you're growing by a factor of 1.75, this is the same thing as adding 75%. Once again, you are adding 75%. Think about it this way. If you just grew by a factor of one, then you're not adding anything. You're staying constant. If you grow by 10%, then you're going to be 1.1 times as large. If you grow by 200% then you're going to be two times as large. So this right over here? This right over here is, is, is. If you, let me be very
careful, what I just said. I think I just mistake that. If you grow by 200%, you are going to be three times as large as you were before. One is constant, and then another 200% would be another two-fold so that would make you three times as large. Don't want to confuse you. My brain recognized that I said something weird right at that end. All right, hopefully you enjoyed that.