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1.6. Practice Problems
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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.
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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.
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Create a free accountFor 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!
Overview
Short Summary
This section presents practice problems focused on exponential growth and decay concepts.
Medium Summary
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.
Detailed Summary
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.
Audio Book
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Create a free account📝 Set 1 – Exponential Growth
- A town has 1,000 people and grows at 4% per year. Find the population after 10 years.
- A tree grows 5% taller each year. If it is 2 meters tall now, how tall will it be in 6 years.
Detailed Explanation
This set of practice problems focuses on exponential growth, where the quantity increases by a specific percentage over time. The first problem involves a town population increasing at an annual growth rate of 4%. To find the population after 10 years, we can use the formula for exponential growth:
𝑦 = 𝑎(1 + 𝑟)𝑡
where:
- 𝑎 is the initial population (1,000),
- 𝑟 is the growth rate as a decimal (0.04),
- 𝑡 is the number of years (10).
For the second problem, we need to calculate the height of a tree that grows at a rate of 5% annually. We apply the same formula to find the height after 6 years.
Examples & Analogies
Imagine a savings account that gains interest each year. If you start with $1,000 and your money increases by 4% annually, you can think of this as the town's population growing. Just like your savings account, the population increases each year, and after a decade, it will be significantly larger than when it started!
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Create a free account📝 Set 2 – Exponential Decay 3. A phone battery loses 20% of its charge every hour. If the battery starts at 100%, how much charge is left after 3 hours? 4. A painting's value depreciates at 6% per year. If it’s worth $10,000 today, what will it be worth in 7 years?
Detailed Explanation
This set of practice problems focuses on exponential decay, where the quantity decreases by a specific percentage over time. The first problem deals with a phone battery that loses 20% of its charge each hour. We can use the exponential decay formula:
𝑦 = 𝑎(1 - 𝑟)𝑡
where:
- 𝑎 is the initial charge (100%),
- 𝑟 is the decay rate as a decimal (0.20),
- 𝑡 is the number of hours (3).
The second problem looks at a painting's worth, which depreciates at 6% per year. Using the same decay formula, we can determine what the painting will be valued at after 7 years.
Examples & Analogies
Think of a car's value over time. Just like a painting loses its value each year, so does the car. If you buy a car for $20,000 and it depreciates by 15% each year, its worth will keep getting lower - similar to how your phone battery drains after each use. Both situations highlight how things lose value or charge over time!
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Exponential Growth: A consistent percentage increase in a quantity over time.
Exponential Decay: A consistent percentage decrease in a quantity over time.
Base (b): Determines whether growth or decay is occurring based on its value.
Examples
Memory Aids
Interactive tools to help you remember key concepts
Stories
Memory Tools
Flash Cards
Glossary
Exponential Growth
A situation in which a quantity increases at a constant percentage over time.
Exponential Decay
A process in which a quantity decreases at a constant percentage over time.
Initial Value (a)
The starting quantity before any growth or decay occurs.
Growth Rate (r)
The percentage increase of a quantity in exponential growth.
Decay Rate (r)
The percentage decrease of a quantity in exponential decay.
Time (t)
The period over which growth or decay is measured.