Aradioactive material has a decay rate proportional to the amount of radioactive material present at that time, with a proportionality factor of 2 per unit time. write a differential equation of the form pf(p), which models this situation, where p is the amount or radioactive material micrograms as function of time

a). P' = -2P

b). P' = -2(P + 4t)

Step-by-step explanation:

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a). A radioactive material has a decay rate P', is proportional to the amount 'P' of radioactive material present at that time.

Relation that represents this phenomenon will be,

P' ∝ -P

P' = -2P [2 is the proportionality factor]

Here negative notation represents the decay of the substance and P' represents the rate of decay.

b). Now additional amount of  radioactive element is added at the rate of 4 micrograms per unit time.

If the radioactive element has been added for the time 't' then amount of radioactive element after time 't' = (P + 4t)

Now we know decay rate is proportional to the amount of radioactive material at that time.

Therefore, P' ∝ -(P + 4t)

P' = -2(P + 4t)