Fungsi Eksponen Pertumbuhan dan Peluruhan

Summary

This video tutorial explains exponential functions related to growth and decay, with practical applications in population growth, depreciation, and radioactive decay. The presenter demonstrates how to model exponential growth using the formula Y = Y0(1 + R)^X, with examples like a 2% population increase and compound interest. For decay, the formula Y = Y0(1 - R)^X is used, exemplified by car depreciation and radioactive iodine half-life. The video includes detailed explanations, example calculations, and step-by-step guides to help viewers understand and apply these concepts in real-life scenarios.

Takeaways

• 😀 Exponential growth is modeled by the formula Y = Y₀ × (1 + r)^x, where Y₀ is the initial value, r is the growth rate, and x is the number of periods.
• 😀 Exponential decay is modeled by the formula Y = Y₀ × (1 - r)^x, where r is the decay rate, and the value decreases over time.
• 😀 The population growth example uses a 2% growth rate to model Indonesia’s population increase from 200 million in 2010, with a formula of Y = 200 × 1.02^x.
• 😀 For compound interest, the formula Y = Y₀ × (1 + r)^x is used to model the growth of a deposit over time, as shown by the 10 million deposit growing with a 5% annual interest rate.
• 😀 The depreciation of a car’s value is modeled using the formula Y = Y₀ × (1 - r)^x, with an 8% annual depreciation resulting in a decrease in value over time.
• 😀 Half-life decay, such as in radioactive substances, is modeled using Y = Y₀ × (0.5)^x, where the amount halves every set time period.
• 😀 To model insect population growth that doubles every month, use the formula Y = Y₀ × 2^x, where x represents the number of months.
• 😀 For population growth problems, carefully calculate the number of years (x) by finding the difference in years between the start and end dates.
• 😀 When calculating decay or growth over time, use a calculator for accurate results, especially when raising a number to a power.
• 😀 The key difference between growth and decay is the sign in the formula: growth uses (1 + r), while decay uses (1 - r) for decreasing values.

Q & A

What is the general formula for exponential growth?

The general formula for exponential growth is Y = Y₀ × (1 + R)^X, where Y is the value after X periods, Y₀ is the initial value, R is the growth rate per period, and X is the number of periods.

What is the difference between exponential growth and exponential decay?

Exponential growth uses the formula Y = Y₀ × (1 + R)^X, where the value increases over time, while exponential decay uses Y = Y₀ × (1 - R)^X, where the value decreases over time.

How is the population growth of Indonesia modeled in the script?

The population growth of Indonesia in 2010, with an initial population of 200 million and a growth rate of 2%, is modeled by the equation Y = 200 × (1.02)^X, where X represents the number of years after 2010.

How do you calculate the population of Indonesia in 2021 based on the growth model?

To calculate the population in 2021, substitute X = 11 (the number of years between 2010 and 2021) into the equation Y = 200 × (1.02)^11. This gives a population of approximately 248.66 million.

Why is the formula for exponential decay different from that of exponential growth?

The formula for exponential decay is Y = Y₀ × (1 - R)^X, where the value decreases over time, while exponential growth uses Y = Y₀ × (1 + R)^X, where the value increases. The key difference is the use of a minus sign for decay.

How is the depreciation of a car's price modeled in the script?

The depreciation of a car's price, starting at 300 million with an 8% annual depreciation rate, is modeled by the formula Y = 300 × (1 - 0.08)^X, where X is the number of years. After 3 years, the car’s value is approximately 233.61 million.

What does the term 'half-life' mean in the context of radioactive decay?

The term 'half-life' refers to the time it takes for half of a substance to decay. In the script, the half-life of iodine-131 is 12 days, meaning the substance reduces by half every 12 days.

How do you calculate the remaining amount of iodine-131 in the body after 48 days?

To calculate the remaining iodine-131 after 48 days, we divide 48 by the half-life period of 12 days, which gives 4 periods. Using the formula Y = Y₀ × (0.5)^X, where Y₀ is 24 grams, the remaining iodine after 48 days is 1.5 grams.

What is the formula for calculating compound interest, and how is it applied in the script?

The formula for compound interest is Y = Y₀ × (1 + R)^X, where Y₀ is the initial deposit, R is the interest rate, and X is the number of periods. In the script, Doni deposits 10 million at a 5% annual interest rate for 6 years, resulting in a final amount of 13.4 million.

How is the insect population modeled in the script, and what happens after 5 months?

The insect population doubles every month, so the formula is Y = 8 × 2^X, where 8 is the initial population and X is the number of months. After 5 months, the population reaches 256 insects.