Write and solve the differential equation that models the following verbal state

Question

Write and solve the differential equation that models the following verbal statement. Evaluate the solution at the specified value of the independent variable. The rate of change of P is proportional to P. When t = 0, P = 2,000 and when t = 1, P = 1,700. What is the value of P when t = 6? Write the differential equation. (Use k for the constant of proportionality.) dP/dt= kP Correct: Your answer is correct. Solve the differential equation. Evaluate the solution at the specified value of the independent variable. (Round your answer to three decimal place thanks in advance!

Explanation / Answer

dP/dt =kP

dP/P= kdt

integrate on both sides

dP/P= kdt

lnP=kt +c

P=ekt +c

P=Cekt

When t = 0, P = 2,000

2000=Ce0

C=2000

P=2000ekt

when t = 1, P = 1,700

1700=2000ek*1

ek=1700/2000

k=ln(1700/2000)

k=-0.16252

P=2000e-0.16252t

for  the value of P when t = 6

P=2000e-0.16252*6=754.299

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