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?

Question

Mathematics

Isabel

20 December, 06:43

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?

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Answers (1)

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Jaidyn Figueroa · 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.