3. Write the exponential growth function to model the following situation:
A population of 422, 000 increases by 12% each year.
4. At t = 0 there are 50 grams of a radioactive isotope. The isotope has a half-life of 16 minutes. Use the exponential decay model to write the amount A as a function of time t.
3. To write the exponential growth function for this situation, we can use the formula:
A(t) = P(1 + r)^t
where A(t) is the final population after time t, P is the initial population, r is the growth rate per time period, and t is the number of time periods.
Given the following information:
Initial population (P) = 422,000
Growth rate (r) = 12% = 0.12
Time (t) is not specified.
Plug in these values into the formula and the exponential growth function would be:
A(t) = 422,000(1 + 0.12)^t
4. To write the exponential decay model for this situation, we can use the formula:
A(t) = P(1/2)^(t/h)
where A(t) is the amount of the isotope at time t, P is the initial amount of the isotope, t is the time, and h is the half-life of the isotope.
Given the following information:
Initial amount (P) = 50 grams
Half-life (h) = 16 minutes
Time (t) is not specified.
Plug in these values into the formula and the exponential decay model would be:
A(t) = 50(1/2)^(t/16)
To write an exponential growth function, we start with the formula:
A = P(1 + r)^t
where:
A represents the final amount or population,
P is the initial amount or population,
r is the growth rate (expressed as a decimal),
and t is the number of time periods.
1. For the population growth problem:
Given:
Initial population (P) = 422,000
Growth rate (r) = 12% = 0.12 per year
Therefore, the exponential growth function is:
A = 422,000(1 + 0.12)^t
2. For the radioactive decay problem:
Given:
Initial amount (P) = 50 grams
Half-life (t1/2) = 16 minutes
The formula for exponential decay is:
A = P(1/2)^(t/t1/2)
To use this formula, we need to convert the half-life from minutes to the relevant time period. If we want to measure it in minutes, then t should also be in minutes. If we prefer using hours, then t should be in hours. It depends on the desired unit of time.
Suppose we want to measure it in minutes, then our formula becomes:
A = 50(1/2)^(t/16)
This equation gives us the amount of the radioactive isotope at any given time (t) in minutes.
p(t) = 422000 * 1.12^t
A(t) = 50(1/2)^(t/16)
Note that every time t grows by 16, A(t) is 1/2 as much.