A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 51 cells.
(a) Find the relative growth rate. (Assume t is measured in hours.)
b)Find an expression for the number of cells after t hours.
(c) Find the number of cells after 8 hours.
(d) Find the rate of growth after 8 hours. (Round your answer to three decimal places.)
(e) When will the population reach 20,000 cells? (Round your answer to two decimal places.)
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To answer these questions, we will use the concept of exponential growth. The growth of a bacterial population can be modeled by the equation N(t) = N0 * e^(rt), where N(t) represents the number of cells at time t, N0 represents the initial number of cells, r represents the relative growth rate, and e is Euler's number, approximately equal to 2.71828.
(a) To find the relative growth rate, we can use the formula r = (ln(N(t)) - ln(N0)) / t, where ln represents the natural logarithm. Using the given information, N(t) = 2N(0) after 20 minutes, and in general, N(t) = 2^(t/20) * N(0) for any t. Substituting these values into the formula, we get:
r = (ln(2^(t/20) * N(0)) - ln(N0)) / t
= (ln(2^(t/20)) + ln(N(0)) - ln(N0)) / t
= (t/20 * ln(2) + ln(N(0)) - ln(N0)) / t
= ln(2) / 20 + ln(N(0))/t - ln(N0)/t
Thus, the relative growth rate is given by r = ln(2)/20.
(b) An expression for the number of cells after t hours can be obtained using the formula N(t) = N0 * e^(rt):
N(t) = 51 * e^((ln(2)/20) * t)
(c) To find the number of cells after 8 hours, we substitute t = 8 into the equation from part (b):
N(8) = 51 * e^((ln(2)/20) * 8)
(d) The rate of growth after 8 hours can be found by taking the derivative of the equation N(t) = 51 * e^((ln(2)/20) * t) with respect to time t and evaluating it at t = 8:
Rate of growth after 8 hours = dN(t)/dt = (51 * (ln(2)/20) * e^((ln(2)/20) * t)) evaluated at t = 8.
(e) To find when the population will reach 20,000 cells, we set N(t) = 20,000 and solve for t:
20,000 = 51 * e^((ln(2)/20) * t)
We can use logarithms to solve this equation. Taking the natural logarithm of both sides gives:
ln(20,000) = ln(51 * e^((ln(2)/20) * t))
Simplifying further:
ln(20,000) = ln(51) + (ln(2)/20) * t
Now, isolate t:
t = (ln(20,000) - ln(51)) / (ln(2)/20)
Using a calculator, we can find the value of t approximately.
(a) To find the relative growth rate, we need to determine how many times the initial population doubles during the given time period.
Since one cell divides into two cells every 20 minutes, the relative growth rate (RGR) can be calculated as:
RGR = 2^(1/20) - 1
(b) The expression for the number of cells after t hours can be calculated using the formula for exponential growth:
N(t) = N0 * (1 + RGR)^(t/Δt)
Where N(t) is the number of cells after t hours, N0 is the initial population, RGR is the relative growth rate, and Δt is the time interval for one doubling (20 minutes, or 1/3 hour).
So, the expression for the number of cells after t hours is:
N(t) = 51 * (1 + RGR)^(t/(1/3))
(c) To find the number of cells after 8 hours, substitute t = 8 into the expression derived in step (b):
N(8) = 51 * (1 + RGR)^(8/(1/3))
(d) To find the rate of growth after 8 hours, we need to find the derivative of the number of cells with respect to time at t = 8. The rate of growth can be calculated as:
Rate of growth = dN(t)/dt | t = 8
Differentiating the expression derived in step (b) with respect to t and then evaluating at t = 8:
Rate of growth = (1 + RGR)^(t/(1/3)) * 51 * ln(1 + RGR) / (1/3) | t = 8
(e) To find when the population reaches 20,000 cells, we need to solve the equation N(t) = 20000 with respect to t. Using the expression derived in step (b):
51 * (1 + RGR)^(t/(1/3)) = 20000
Solving for t will give the time required for the population to reach 20,000 cells.