Question

The decreasing function M represents the mass of a sample of a radioactive element at time t. The rate of change of the sample’s mass is proportional to the mass of the sample. Which of the following differential equations could be used to model the relationship between the mass of the sample and the rate of change of the sample’s mass, where k is a constant?A)dM =kt dtb) d - = KM dtc)DMd)dᎷ .

Answer 1

Answer:

$(B) \dfrac{dM}{dt} = kM$

Step-by-step explanation:

Given the mass M(a decreasing function) of a sample of a radioactive element at time t.

The rate of change of the sample’s mass $\dfrac{dM}{dt}$, is proportional to the mass, M of the sample.

This is written as:

$\dfrac{dM}{dt} \propto M$

Introducing the decay constant k,

$\dfrac{dM}{dt} = kM$

This is the equation which model the relationship between the mass of the sample and the rate of change of the sample’s mass.

The other options are therefore invalid.

$(A)\dfrac{dM}{dt} = kt (C)\dfrac{dM}{dt} = \dfrac{k}{t} (D)\dfrac{dM}{dt} = \dfrac{k}{M}$