Exponential Decay and Growth

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What is exponential decay?

Back

Exponential decay is a process where a quantity decreases at a rate proportional to its current value, often represented by the formula y = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

Front

What is exponential growth?

Back

Exponential growth is a process where a quantity increases at a rate proportional to its current value, often represented by the formula y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

Front

What is the formula for calculating the future value of an investment with exponential growth?

Back

The formula is y = a(1 + r)^t, where 'y' is the future value, 'a' is the initial investment, 'r' is the growth rate, and 't' is the time in years.

Front

What is the formula for calculating the future value of an asset with exponential decay?

Back

The formula is y = a(1 - r)^t, where 'y' is the future value, 'a' is the initial value, 'r' is the decay rate, and 't' is the time in years.

Front

If a population of 1000 decreases by 10% each year, what will be its population after 2 years?

Back

The population after 2 years will be approximately 810.

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What does a decay rate of 5% mean in terms of exponential decay?

Back

A decay rate of 5% means that each year, the quantity decreases to 95% of its value from the previous year.

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How do you determine the half-life of a substance?

Back

The half-life is the time required for a quantity to reduce to half its initial value, often calculated using the formula t_half = ln(2)/r, where 'r' is the decay rate.

Front

What is the significance of the base in exponential functions?

Back

The base in exponential functions determines the rate of growth or decay; a base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.

Front

If a car's value is represented by v(t) = 20,000(0.85)^t, what is the depreciation rate?

Back

The depreciation rate is 15%.

Front

What is the relationship between exponential functions and logarithms?

Back

Exponential functions can be inverted using logarithms; for example, if y = a^x, then x = log_a(y).