Suppose that a population of bacteria triples every hour and starts with 700 bacteria.
(a) Find an expression for the number n of bacteria after t hours.
n(t) = ?
(b) Estimate the rate of growth of the bacteria population after 1.5 hours. (Round your answer to the nearest hundred.)
n'(1.5) = ?
A population starts with 1000 individuals and triples every 80 years.
a. give an exponential model for this situation
b. what is the size of the population after 100 years?
(a) To find an expression for the number of bacteria after t hours, we can use the formula for exponential growth:
n(t) = initial population * growth rate^time
Given that the population triples every hour, the growth rate is 3 (since 3 * initial population = triple the initial population).
Therefore, the expression for the number of bacteria after t hours is:
n(t) = 700 * (3^t)
(b) To estimate the rate of growth of the bacteria population after 1.5 hours, we need to find the derivative of the function n(t).
Taking the derivative of n(t) = 700 * (3^t) with respect to t gives us:
n'(t) = 700 * ln(3) * (3^t)
To find the rate of growth after 1.5 hours, we substitute t = 1.5 into the derivative expression:
n'(1.5) = 700 * ln(3) * (3^1.5)
Calculating this expression will give us the approximate rate of growth of the bacteria population after 1.5 hours.
To find the expression for the number of bacteria after t hours, we can use the fact that the bacteria triples every hour. Starting with 700 bacteria, after the first hour, the population would triple to 3 * 700 = 2100.
Therefore, we can write the expression as:
n(t) = 700 * 3^t
For part (b), we need to estimate the rate of growth of the bacteria population after 1.5 hours. To do this, we can find the derivative of the expression n(t) with respect to t and evaluate it at t = 1.5.
Taking the derivative of n(t) = 700 * 3^t with respect to t, we get:
n'(t) = 700 * ln(3) * 3^t
Now, substituting t = 1.5 into the derivative expression, we have:
n'(1.5) = 700 * ln(3) * 3^1.5
To estimate this value, we need to compute ln(3) and 3^1.5:
ln(3) ≈ 1.10
3^1.5 ≈ 5.20
Now, substituting these values into the expression for n'(1.5), we get:
n'(1.5) ≈ 700 * 1.10 * 5.20
Evaluating this expression, we find that n'(1.5) ≈ 3982.00. Rounded to the nearest hundred, the estimated rate of growth of the bacteria population after 1.5 hours is 3900.