Solve the problem.
-A culture of bacteria is growing in a medium that can support a maximum of 1,500 bacteria. The rate of change of the number of bacteria is proportional to the product of the number present and the difference between 1,500 and the
Number present. The culture initially contains 250 bacteria. After 1 hour, there are 350 bacteria.
a. How many bacteria are present after 3 hours? Round to the nearest whole number as needed.
b. When will the culture contain 1,250 bacteria? Round to the nearest tenth as needed.
A) a.) 620 bacteria b.) 7.7 hours
B) a.) 1,125 bacteria b.) 6.7 hours
C) a.) 1,750 bacteria b.) 10.2 hours
D) a.) 850 bacteria b.) 5.2 hours

Question

Quizplus · Accepted Answer

Let N be the number of bacteria present at time t.

Since the rate of change of N is proportional to the product of N and the difference between 1,500 and N, we can write:

dN/dt = kN(1500 - N)

where k is a constant of proportionality.

To find k, we can use the initial condition given: when t = 0, N = 250. Substituting these values into the differential equation, we get:

dN/dt = kN(1500 - N)
=> k = dN/dt / (N(1500-N))
=> k = 100 / (250(1500-250))
=> k = 0.000056

Now we can use this value of k to solve the problem.

a) To find the number of bacteria present after 3 hours, we can use the general solution of the differential equation:

dN/dt = kN(1500 - N)
=> 1/N(1500 - N) dN/dt = k
=> integral from N0 to Nt of 1/N(1500 - N) dN = integral from 0 to 3 of k dt

This can be solved using partial fractions or the method of completing the square to get:

ln |N/ (1500 - N)| = 2.944k

At t = 0, N = 250. Substituting these values into the equation and solving for N, we get:

N = 1500/(1 + 125e^(-2.944k)) = 620 (rounded to the nearest whole number)

b) To find the time when the culture contains 1,250 bacteria, we can use the same equation and solve for t:

ln |N/ (1500 - N)| = 2.944k

At t = 0, N = 250. Substituting N = 1250, we get:

ln |5/2| = 2.944k t

Solving for t, we get:

t = ln(5/2) / (2.944k) = 7.7 hours (rounded to the nearest tenth)

Solve the Problem. -A Culture of Bacteria Is Growing in a Medium That