a) The population size of a country is increasing at a rate
of 4% per year. How long does it take until the country has doubled
its population size?
b) The number of new flu cases is decreasing at a rate of 5% per week.
After how much time will the number of new flu cases reach a quarter
of its current level?
1.04^n =2
n log 1.04 = log 2
n = log 2 / log 1.04
second one same way
0.95^n = 0.25
a) To determine how long it takes for a country's population to double, we need to use the concept of exponential growth. In this case, the population is increasing at a rate of 4% per year.
To find the time it takes for the population to double, we can use the formula for exponential growth:
Final Population = Initial Population * (1 + Growth Rate)^Time
Let's denote the initial population as P0, the final population as P, and the growth rate as r. In this case, P = 2 * P0 (as we want the population to double) and r = 4% (or 0.04 as a decimal). Substituting these values into the formula, we get:
2 * P0 = P0 * (1 + 0.04)^Time
Canceling out P0 on both sides, we have:
2 = (1 + 0.04)^Time
To solve for Time, we need to take the logarithm of both sides of the equation. Let's assume we use the natural logarithm (ln):
ln(2) = ln((1 + 0.04)^Time)
Using the property of logarithms, we can bring the exponent down:
ln(2) = Time * ln(1 + 0.04)
Finally, we can solve for Time by dividing both sides of the equation by ln(1 + 0.04):
Time = ln(2) / ln(1 + 0.04)
Using a calculator, we can calculate the value of Time.
b) To find out how much time it takes for the number of new flu cases to reach a quarter of its current level, we need to use the concept of exponential decay. In this case, the number of new flu cases is decreasing at a rate of 5% per week.
Similarly to part a), let's denote the initial number of flu cases as C0, the final number of flu cases as C, and the decay rate as r. In this case, C = C0/4 (as we want the number of cases to decrease to a quarter) and r = -5% (or -0.05 as a decimal). We need to set up a formula using exponential decay:
Final Number of Cases = Initial Number of Cases * (1 + Decay Rate)^Time
Substituting the given values, we get:
C0/4 = C0 * (1 - 0.05)^Time
Canceling out C0 on both sides, we have:
1/4 = (0.95)^Time
To solve for Time, we need to take the logarithm of both sides of the equation:
ln(1/4) = ln((0.95)^Time)
Using the property of logarithms, we bring the exponent down:
ln(1/4) = Time * ln(0.95)
Finally, we can solve for Time by dividing both sides of the equation by ln(0.95):
Time = ln(1/4) / ln(0.95)
Again, using a calculator, we can find the value of Time.