Write and find the general solution of the differential equation that models the 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 = 5000, and when t = 1, P = 4750. What is the value of P when t = 5?

Step-by-step explanation:

Rate of change of P is proportional to P

i.e dP/dt ∝ P

Let k be constant of proportionality

Then,

dP/dt =kP

Using variable separation

Then,

1/P dP=kdt

Integrating both side

∫1/P dP= ∫ kdt

InP=kt+c

Taking exponential of both side

P=exp(kt+c)

P=exp(kt)exp(c)

exp(c) is also a constant let say A

Then,

P=Aexp(kt)

Using the initial conditions

At t=0, P=5000

5000=Aexp(0)

Then A=5000

Therefore the equation becomes

P=5000exp(kt)

At t=1, P=4750, let substitute this

4750=5000exp(k×1)

Divide both side by 5000

0.95=exp(k)

Take In of both side

In0.95=k

k=-0.0513

The new equation becomes

P=5000exp(-0.0513t)

Now let find P at t=5

P=5000 exp(-0.0513×5)

P=5000 ×exp(-0.25647)

P=5000 × 0.77378

P=3868.89.

P=3869. Approximately