Part 1: How are exponential growth and decay present in the real world? Give at least 2 examples for exponential growth and 2 examples of exponential decay.
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temperature change in a hot cup of tea as it cools.
Body temperature of a dead person
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Exponential growth and decay are observed in various aspects of the real world. Let's explore two examples of exponential growth and two examples of exponential decay:
Exponential growth:
1. Population growth: Over time, populations can experience exponential growth. For example, imagine a species of bacteria introduced into a nutrient-rich environment. Initially, there may only be a few bacteria, but since they reproduce rapidly, the population can quickly grow exponentially.
To calculate the growth rate, you can use the formula P = P0 * e^(rt), where P is the population size at a given time, P0 is the initial population size, e is Euler's number (approximately 2.718), r is the growth rate, and t is the time.
2. Compound interest: When money is invested over time, it can experience exponential growth through compound interest. For instance, depositing money into a savings account that yields a fixed interest rate. As time goes on, the interest earned is added to the initial amount, leading to exponential growth of the account balance.
To calculate compound interest, you can use the formula A = P(1 + r/n)^(nt), where A is the final amount, P is the principal (initial amount), r is the interest rate expressed as a decimal, n is the number of times interest is compounded per time period, and t is the time in years.
Exponential decay:
1. Radioactive decay: Many radioactive elements undergo exponential decay, gradually decreasing their quantity over time. For example, Carbon-14 dating uses the decay of Carbon-14 isotopes in organic materials to estimate their age. As the Carbon-14 isotopes decay, the amount remaining decreases exponentially.
To calculate the decay rate, you can use the formula N(t) = N0 * e^(-λt), where N(t) is the remaining quantity at time t, N0 is the initial quantity, e is Euler's number, λ is the decay constant, and t is the time.
2. Drug concentration: When drugs are metabolized or excreted from the body, their concentration can follow an exponential decay pattern. This decay is often described by the half-life of the drug, which is the time it takes for the drug concentration to decrease by half. With each half-life, the remaining drug concentration halves again.
To calculate the remaining drug concentration, you can use the formula C(t) = C0 * 0.5^(t/h), where C(t) is the remaining concentration at time t, C0 is the initial concentration, t is the time elapsed, and h is the half-life of the drug.
By understanding these examples and the associated formulas, you can better grasp the concept of exponential growth and decay in the real world.