Question

Extreme heat applied to a colony of microorganisms causes the size P of the colony, measured in grams, to decrease according to the exponential decay model dP/dt=−0.4P, where the time t is measured in hours. The size Q of a second colony of microorganisms, also measured in grams, decreases at the constant rate of 1 gram per hour according to the linear model dQ/dt=−1. If at time t=0 the first colony has size P(0)=2 and the second colony has size Q(0)=3, at what time will both colonies have the same size?

Answer 1

Step-by-step explanation:

dP/dt = -0.4P

Separate the variables and integrate.

dP/P = -0.4 dt

ln|P| = -0.4t + C

P = Ce^(-0.4t)

Use initial condition to find C.

2 = C^(0)

C = 2

P = 2e^(-0.4t)

dQ/dt = -1

Integrate.

Q = -t + C

Use initial condition to find C.

3 = 0 + C

C = 3

Q = -t + 3

Set the two equal.

2e(-0.4t) = -t + 3

Use a calculator to find t.

t = 2.156

The populations have the same size after 2.156 hours.