A bacterial population starts at 500 and doubles every four days. Find a formula for the population after t days and find the percentage rate of change in population.
population= Original*(2^(t/5))
rate of change
dpopulation/dt= t/5 * ln2* 2^(t/5) * original
for percentate, then percent= 100*dpopulation/dt / original= 100*t/5*ln2*2^t/5
To find a formula for the population after t days, we need to determine the growth rate of the bacterial population.
Given that the bacterial population doubles every four days, we know that the growth rate is 2.
Let P(t) represent the population at time t.
We also know that the initial population (P(0)) is 500.
Therefore, the formula for the population after t days can be expressed as:
P(t) = P(0) * (growth rate)^n
Substituting the given values, we have:
P(t) = 500 * 2^(t/4)
To find the percentage rate of change in population, we can calculate the derivative of the population function with respect to time and evaluate it at a specific time.
Taking the derivative of P(t) with respect to t, we get:
dP(t)/dt = (500 * ln(2))/4 * 2^(t/4)
To find the percentage rate of change, we divide the derivative by the population at time t and multiply by 100:
Rate of change = (dP(t)/dt) / P(t) * 100
Substituting the values, we have:
Rate of change = (500 * ln(2))/4 * 2^(t/4) / (500 * 2^(t/4)) * 100
Simplifying this expression, we get:
Rate of change = (ln(2)/4) * 100
Therefore, the percentage rate of change in population is approximately 17.3%.
To find a formula for the population after t days, we need to determine the growth pattern of the bacterial population. We know that the bacterial population doubles every four days.
Let's analyze the growth pattern step by step:
After 0 days, the population is 500.
After 4 days, the population doubles to 1000.
After 8 days, the population doubles again to 2000.
After 12 days, the population doubles to 4000.
Based on these observations, we can conclude that the population doubles every 4 days. Therefore, the growth rate can be expressed as 2^(t/4).
To find the formula for the population after t days, we multiply the initial population (500) by the growth rate:
Population = 500 * 2^(t/4).
Now let's find the percentage rate of change in population. The percentage rate of change is calculated by finding the derivative of the population equation with respect to time (t) and then multiplying it by 100 to convert it to a percentage.
Derivative of the population equation with respect to t is given by:
dP/dt = (500 * ln(2) * 2^(t/4))/4
To find the percentage rate of change, we multiply the derivative by 100 and divide it by the current population:
Percentage Rate of Change = (dP/dt * 100) / Population
Substituting the values, we get:
Percentage Rate of Change = [(500 * ln(2) * 2^(t/4))/4 * 100] / (500 * 2^(t/4))
Simplifying further, the percentage rate of change formula becomes:
Percentage Rate of Change = ln(2) * 25
Therefore, the formula for the population after t days is: Population = 500 * 2^(t/4).
The percentage rate of change in population is approximately equal to ln(2) * 25.