Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Today, we are going to delve into exponential growth. Remember, exponential growth means a quantity increases by a fixed percentage over regular intervals. Can anyone tell me the formula for calculating exponential growth?
Is it y = a(1 + r)^t?
Exactly! Here, 'a' represents your initial amount, 'r' is the growth rate as a decimal, and 't' is time. Let’s try an example together. If a town has 1,000 people and grows at 4% per year, how would we find its population after 10 years?
I think we need to calculate y using the formula you just mentioned!
Correct! You would substitute a = 1000, r = 0.04, and t = 10 into the formula. What do you get?
I got approximately 1,480 people!
Great job! And that’s how we apply the exponential growth formula.
Next, we’ll talk about exponential decay. When do we encounter decay in real life?
Maybe when a car or something loses value over time?
Exactly! Exponential decay happens when a quantity decreases by a fixed percentage. The formula is y = a(1 - r)^t. Let's apply it! If a phone battery starts at 100% and loses 20% of its charge every hour, what would be the charge left after 3 hours?
So, a = 100, r = 0.20, and t = 3, right? I can use those in the formula.
Yes! Substituting those values, what do you find?
I calculated about 51.2% left after 3 hours!
Fantastic! You've just illustrated exponential decay effectively.
For example, in ecology, populations can grow while individual health may decline! How about we work on a new example together?
That sounds interesting! What would we do?
Let’s say a grapevine grows, and it initially has a growth rate of 10% per year, but due to pests, it loses 5% of its yield annually. Can someone attempt to set it up using our formulas for both growth and decay?
We would use growth for the initial year then adjust that value with the decay formula the next.
Exactly! You would calculate the growth for the first year, then apply the decay for the next. It’s a perfect mix of what we’ve learned!
In growth, we rise, in decay, we fall! Percentages guide us, we use the growth call!
Imagine a tree that grows every year, quickly and broadly, but when pests appear, its leaves disappear, representing decay.
GRAPES: Growth Rate, Amount, Period, Exponential (for growth); decrease Rate, Amount, Period (for decay).
FAG: For growth: Formula = a(1 + g)^t; For decay: Formula = a(1 - d)^t.
Term
Exponential Growth Formula
Click to flip
Term
Exponential Decay Formula
Click to flip
A situation in which a quantity increases at a constant percentage over time.
A process in which a quantity decreases at a constant percentage over time.
The starting quantity before any growth or decay occurs.
The percentage increase of a quantity in exponential growth.
The percentage decrease of a quantity in exponential decay.
The period over which growth or decay is measured.
This section presents practice problems focused on exponential growth and decay concepts.
The practice problems designed for exponential growth and decay cover various scenarios such as population growth and financial depreciation. These problems encourage the application of formulas in real-world contexts.
The section 'Practice Problems' consists of two sets of exercises: one focusing on exponential growth and another on exponential decay. The exercises require students to apply the specific formulas for exponential growth (y = a(1+r)^t) and decay (y = a(1−r)^t) to solve problems related to real-world situations, such as population changes and depreciation of asset values. By tackling these practice problems, students gain familiarity with identifying variables in contexts, calculating future values, and interpreting the results within a mathematical framework.