the number of bacteria in a certain population is predicted to increase according to a continuous exponential growth model at a relative rate of 6% per hour. suppose that a sample culture has an initial population of 94 bacteria. find the predicted population after four hours.
p = 94 e^(.06 * 4)
94 * .06 = the first hour
times that number by 4
To find the predicted population after four hours using the continuous exponential growth model, we can use the formula:
P(t) = P0 * e^(rt)
where:
P(t) is the predicted population after time t
P0 is the initial population
e is the base of the natural logarithm (approximately 2.71828)
r is the relative rate of growth per time unit
t is the time elapsed
Given:
P0 = 94 (initial population)
r = 6% per hour (relative rate of growth)
Substituting these values into the formula, we have:
P(4) = 94 * e^(0.06 * 4)
To calculate this, we need the value of e^(0.24). Using a calculator:
e^(0.24) approximately equals 1.271
Therefore,
P(4) = 94 * 1.271
P(4) = 119.534
So, the predicted population after four hours is approximately 120 bacteria.
To find the predicted population after four hours, we can use the continuous exponential growth formula:
P(t) = P0 * e^(rt)
Where:
P(t) = the predicted population at time t
P0 = the initial population
r = the relative growth rate (expressed as a decimal)
t = time in hours
e = Euler's number (approximately 2.71828)
In this case, the initial population (P0) is 94 bacteria and the relative growth rate (r) is 6% per hour, which can be expressed as 0.06. Now we can substitute these values into the formula to find the predicted population after four hours.
P(4) = 94 * e^(0.06 * 4)
To calculate this, we need to use the value of e raised to the power of (0.06 * 4), and then multiply it by the initial population.
Step 1: Calculate e^(0.06 * 4)
Step 2: Multiply the result by 94
Let's calculate it step by step:
Step 1: Calculate e^(0.06 * 4)
e^(0.06 * 4) ≈ 1.262
Step 2: Multiply the result by 94
P(4) = 94 * 1.262 ≈ 118.628
Therefore, the predicted population after four hours is approximately 118.628 bacteria.