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A short class with Sarah and her students. The other ways are under More.
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Today, we're going to explore how we can use the compound interest formula to predict population growth. Can anyone tell me why this might be important?
It helps us understand how many people will be in our city in the future!
Exactly! So, if a city's population was 20,000 in 1997 and grows at 5% each year, how would we calculate its population in 2000?
We would use the compound interest formula: Population in future = P * (1 + r)^n.
Great! So we substitute P = 20,000, r = 0.05, and n = 3 years. What does that give us?
It should be 20,000 * (1.05)^3, which is 20,000 times roughly 1.157625, right?
That's right! And what do you find?
The population would be approximately 23,153.
Excellent work! This demonstrates how we can model population growth effectively.
Shifting gears, let’s talk about bacteria. If we start with 500,000 bacteria and they grow at a rate of 2.5% per hour, how can we predict how many there will be in 2 hours?
We would apply the same formula as before, but for a different context!
Correct! If we let P = 500,000, r = 0.025, and n = 2, how would it look?
It would be 500,000 * (1 + 0.025)^2.
And what does that calculate to?
That’s approximately 500,000 * 1.050625, which is around 525,313!
Great job! Bacteria can multiply quickly, and this formula helps us anticipate those numbers.
Let’s discuss depreciation. If you buy a TV for 21,000 and it depreciates by 5% annually, how can we find out its value next year?
We calculate 5% of 21,000, right?
Correct! What is that?
That’s 1,050. So the value will be 21,000 - 1,050, which equals 19,950.
Excellent! We could also write it using our formula: Value = P * (1 - r). How does that look?
So it would be 21,000 * (1 - 0.05), giving 21,000 * 0.95?
Exactly! You’re all catching on! Key point: understanding depreciation is as important as understanding growth.
It is calculated on the principal plus accumulated interest, showing true financial growth.
Often modeled using the compound interest formula to predict future populations.
Can also use compound-like calculations to determine future value reductions of assets.
Example 1: Calculating future population using the formula: 20,000 * (1 + 0.05)^3 = 23,153.
Example 2: Bacteria propagation after two hours would result in a total of approximately 525,313 from an initial population of 500,000 growing at 2.5%.
Example 3: A TV’s depreciation from 21,000 to 19,950 after one year with a 5% depreciation rate.
Values grow fast, and numbers last, with compound interest, you'll have a blast!
Once there was a town that wanted to track its population. Each year they multiplied their citizens, and the numbers flourished just as the trees in spring, thanks to the magic of compound interest.
GIR - Growth In Rates captures the essence of compounding growth!
Term
Compound Growth Formula
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Term
Depreciation
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Interest calculated on the initial principal which also includes all the accumulated interest from previous periods.
The increase in the number of individuals in a population.
Reduction in the value of an asset over time, due to wear and tear or obsolescence.
A growth pattern where the quantity increases by a constant percentage over equal intervals.
This section discusses the practical applications of the compound interest formula in various real-world scenarios.
The section outlines situations where the compound interest formula can be applied, such as population growth, bacterial growth, and the changing value of items due to appreciation or depreciation. Key examples illustrate how to calculate using this formula effectively.
The compound interest formula is not just a theoretical tool; it finds significant application in various real-world scenarios. This section elaborates on several practical situations where the compound interest formula proves to be useful. Among these applications are the calculations related to population growth, bacterial growth rates, and the valuation changes of items due to economic factors.
Understanding the applications of the compound interest formula equips students with a valuable mathematical tool applicable not only in finance but also in biology, economics, and everyday decision-making.