Exponential Growth and Decay Word Problems & Functions - Algebra & Precalculus

The Organic Chemistry Tutor
6 Dec 201612:49
EducationalLearning
32 Likes 10 Comments

TLDRThis video script focuses on solving exponential growth and decay problems through various word problems. It begins with a scenario of a rabbit population on an island growing by 8% annually, using the formula y = a * b^x to calculate that by 2020 there will be approximately 3,172 rabbits. The script then tackles the depreciation of a car over time, using a similar exponential decay formula to determine its value in 2024. It also explores the increase in the value of a home over 13 years and calculates its original purchase price. Finally, the script addresses bacterial growth, first doubling every 20 minutes to find the count in three hours, and then tripling every 15 minutes to determine the time needed to reach 500 million bacteria. The use of logarithms is introduced to solve for time in the latter problem. The video provides a clear understanding of how to apply exponential growth and decay formulas to real-world scenarios.

Takeaways
  • 📈 The formula for exponential growth is y = a * b^x, where a is the initial amount, b is the growth factor (1 + r), and x is the time period.
  • 📉 The formula for exponential decay is y = a * (1 - r)^x, where a is the initial value, r is the decay rate, and x is the time period.
  • 🐰 To find the future population of rabbits, use the growth formula with the initial population, growth rate converted to a decimal, and the number of years as the exponent.
  • 🚗 To calculate the future value of a depreciating asset like a car, use the decay formula with the initial value, decay rate as a decimal, and the time period in years.
  • 🏠 To determine the past purchase price of a property that appreciates, rearrange the growth formula to solve for the initial value (a) using the present value, growth rate, and time period.
  • 🔬 For bacteria growth problems, use the growth formula with the initial count, a growth factor that represents the doubling or tripling rate, and the time period in the appropriate units.
  • ⏱️ The number of times bacteria doubles or triples in an hour can be found by dividing the number of minutes in an hour by the growth interval.
  • 🧮 Use logarithms to solve for the time it takes to reach a certain population size when given the initial count and growth rate.
  • 📊 Exponential growth and decay problems can be solved using a calculator, applying the correct formula, and understanding the relationship between the variables.
  • 💡 It's important to distinguish between growth and decay problems as they use slightly different formulas but the same underlying concept of exponential functions.
  • 📐 In exponential problems, the value represented by 'a' occurs before 'y', which is the value after some time has passed.
Q & A

  • What type of problem is being discussed in the video regarding the rabbit population on the island?

    -The problem is an exponential growth problem because the rabbit population is increasing each year.

  • How is the growth rate of the rabbit population converted into a decimal for the calculation?

    -The growth rate of 8% is converted to a decimal by dividing by 100, which gives 0.08.

  • What is the formula used to calculate the future population of rabbits on the island?

    -The formula used is y = a * b^x, where a is the initial population, b is 1 plus the growth rate, and x is the number of years.

  • How many years pass between 2005 and 2020, according to the video?

    -15 years pass between 2005 and 2020.

  • What is the approximate number of rabbits on the island by 2020, according to the calculation?

    -There will be approximately 3,172 rabbits on the island by 2020.

  • What is the depreciation rate of the car in the second problem?

    -The car depreciates at a rate of 7% each year.

  • How is the depreciation rate used in the exponential decay formula for the car's value?

    -The depreciation rate is used as '1 minus r' in the formula, where r is the depreciation rate expressed as a decimal.

  • What is the expected value of the car in 2024, after nine years of depreciation?

    -The expected value of the car in 2024 is about $20,816.44.

  • In the home value problem, what is the present value of the home in 2015?

    -The present value of the home in 2015 is $225,000.

  • How is the exponential growth formula adjusted to find the original price of the home in 2002?

    -The formula is adjusted by solving for 'a', which represents the original price, using the present value 'y' and the growth factor raised to the power of the number of years.

  • What is the approximate amount John paid for his home in 2002, according to the formula?

    -John paid approximately $135,000 and $129.17 for his home in 2002.

  • How many counts of bacteria are expected in three hours if the sample starts with a thousand and doubles every 20 minutes?

    -There will be 512,000 counts of bacteria in three hours.

  • If a sample contains 100 counts of bacteria and triples every 15 minutes, how many bacteria will there be in one hour?

    -In one hour, there will be 8,100 counts of bacteria.

  • How long will it take for a sample with 100 counts of bacteria to contain 500 million counts if the bacteria triple every 15 minutes?

    -It will take approximately 3.51 hours for the sample to contain 500 million counts of bacteria.

Outlines
00:00
🐇 Exponential Growth of Rabbit Population

This paragraph discusses an exponential growth problem involving the population of rabbits on an island. Starting with 1,000 rabbits in 2005, the population grows by 8% annually. To find the number of rabbits by 2020, the formula y = a * b^x is used, where a is the initial population, b is 1 plus the growth rate (converted to a decimal), and x is the number of years passed. By substituting the values and calculating, the estimated rabbit population in 2020 is found to be approximately 3,172.

05:02
🚗 Exponential Decay in Car Value and Growth in Home Value

The second paragraph covers two financial problems: exponential decay in the value of a car and exponential growth in the value of a home. For the car, which depreciates by 7% annually, the formula y = a * (1 - r)^x is applied to find its value in 2024, resulting in an expected worth of about $20,816.44. For the home purchased by John in 2002, which appreciates by 4% annually, the formula y = a * (1 + r)^x is used to determine the purchase price in 2002, which is calculated to be $135,000.12917.

10:07
🦠 Bacterial Growth Over Time

The third paragraph deals with exponential growth in the context of bacterial counts. Two scenarios are presented: one where bacteria double every 20 minutes and another where they triple every 15 minutes. For the first scenario, the formula y = a * b^(x * n) is used, with b being 2 for doubling, n being 3 (since there are three 20-minute intervals in an hour), and x being the number of hours. This results in 512,000 bacteria after three hours. For the second scenario, b is 3 for tripling, n is 4 (as there are four 15-minute intervals in an hour), and solving for t when y is 500 million gives a time of approximately 3.51 hours for the bacteria to reach that count.

Mindmap
Keywords
💡Exponential Growth
Exponential growth refers to a phenomenon where a quantity increases at a rate proportional to its current value. In the video, it is used to describe the population growth of rabbits on an island, which increases by 8% each year. It is a core concept for understanding how populations or quantities can grow rapidly over time.
💡Exponential Decay
Exponential decay is the process by which a quantity decreases at a rate proportional to its current value. In the video, it is used to calculate the depreciation of a car's value over time. The concept is crucial for understanding how the value of assets can diminish rapidly due to factors like depreciation.
💡Initial Population
The initial population refers to the starting quantity or number of individuals in a given context. In the video, the initial population is the number of rabbits on the island in 2005, which is 1,000. This term is essential as it serves as the base value from which exponential growth or decay calculations begin.
💡Growth Rate
The growth rate is the percentage or multiplier by which a quantity increases or decreases over a specific period. In the video, the growth rate for the rabbit population is 8% per year, and for the car depreciation, it is 7% per year. Understanding the growth rate is key to calculating future values in exponential growth or decay problems.
💡Time Period
The time period is the duration over which a change occurs. In the video, the time period is used to calculate the number of years between 2005 and 2020 for the rabbit population growth, and between 2015 and 2024 for the car's depreciation. The time period is a critical factor in determining the extent of exponential growth or decay.
💡Formula
A formula is a mathematical statement that expresses a relationship between quantities. In the video, formulas are used to calculate exponential growth and decay. For instance, the formula y = a * b^(x) is used, where 'a' is the initial amount, 'b' is the growth or decay factor, and 'x' is the time period. Formulas are essential tools for solving the problems presented in the video.
💡Percentage to Decimal Conversion
Converting a percentage to a decimal is a mathematical operation that involves dividing the percentage by 100. In the video, this conversion is shown when calculating the growth rate (8% becomes 0.08) and the depreciation rate (7% becomes 0.07). This conversion is necessary because mathematical formulas typically require decimal inputs.
💡Depreciation
Depreciation refers to the decrease in the value of an asset over time due to factors such as wear and tear or obsolescence. In the video, the car's value depreciates by 7% each year. Depreciation is an important concept for understanding the financial impact of owning tangible assets over time.
💡Logarithms
Logarithms are mathematical functions that can help solve equations involving exponential growth or decay by reversing the process. In the video, logarithms are used to find the time it takes for the bacteria count to reach 500 million. Logarithms are crucial for solving for the unknown variable 't' in certain exponential problems.
💡Bacteria Growth
Bacteria growth is a biological process where the number of bacteria increases rapidly through cell division. In the video, two examples of bacterial growth are given: one where bacteria double every 20 minutes, and another where they triple every 15 minutes. Understanding bacteria growth is important for modeling and predicting the spread of microorganisms.
💡Present Value
The present value is the current worth of a future sum of money or asset, considering its expected earnings or depreciation. In the video, the present value is used to determine the original cost of a home in 2002, knowing its value in 2015. The concept of present value is essential for making financial decisions and understanding investment growth.
Highlights

The video focuses on solving exponential growth and decay word problems.

A formula for exponential growth is y = a * b^x, where a is the initial population, b is 1 plus the growth rate, and x is time.

To convert a percentage to a decimal, divide by 100. For example, 8% becomes 0.08.

For the rabbit population problem, the calculation is 1000 * 1.08^15, resulting in approximately 3172 rabbits by 2020.

Exponential decay problems use the formula y = a * (1 - r)^t, where r is the decay rate.

A car's value depreciates by 7% each year, calculated as 40,000 * 0.93^9, to find its worth in 2024.

To find the original price of a house that increases by 4% annually, use the formula a = y / (1 + r)^t, where y is the present value.

John's house was worth $135,000 in 2002, calculated using the exponential growth formula with 13 years of growth at 4% annually.

For bacteria doubling every 20 minutes, the formula y = a * b^(n*t) is used, with n being the number of doubling periods per hour.

In three hours, a sample with 1000 bacteria counts will have 512,000 counts, calculated as 1000 * 2^9.

For bacteria tripling every 15 minutes, the formula is y = a * b^(n*t), with n calculated as 60 minutes divided by the doubling interval.

After one hour, a sample with 100 bacteria counts will have 8100 counts, calculated as 100 * 3^4.

To solve for the time it takes to reach a certain bacteria count, logarithms can be used to isolate the variable t.

It will take 3.51 hours for a sample to contain 500 million bacteria counts, using the formula and logarithmic calculation.

The video provides a step-by-step guide to solving exponential growth and decay problems using real-world examples.

Practical applications of exponential growth and decay are demonstrated through problems involving population growth, depreciation, and bacterial growth.

The importance of converting percentages to decimals for calculations is emphasized in the video.

Logarithms are introduced as a tool for solving exponential growth and decay problems involving time.

The video concludes with a comprehensive solution to a complex exponential growth problem involving bacteria counts.

Transcripts

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