Question

The mass m milligrams of a radioactive substance at time t days, is given by:m(t) = Ae^-tkwhere A and k are constants.The initial mass of the substance is 120 milligrams.The mass of the substance after 10 days is 90 milligrams.a)Sketch the graph to show the relationship between tand mfor t≥ 0 days.

Answer 1

Since the initial mass of the substance is 120 milligrams, then:

`m(0)=120.`

Therefore, the y-intercept of the graph of the given function is (0,120), also, evaluating the given function at t=0 we get:

`m(0)=Ae^(-0k)=Ae^0=A\text{.}`

Therefore A=120.

Now, we know that after 10 days, the mass of the substance is 90 milligrams, then the graph of the function passes through the point (10,90), also, evaluating the given function at t=10 we get:

`m(10)=120e^(-10k).`

Therefore:

`90=120e^(-10k)\text{.}`

Dividing the above equation by 120 we get:

`\begin{gathered} (90)/(120)=(120e^(-10k))/(120), \\ (3)/(4)=e^(-10k)\text{.} \end{gathered}`

Applying the natural logarithm to the above equation we get:

`\begin{gathered} \ln ((3)/(4))=\ln (e^(-10k)), \\ \ln ((3)/(4))=-10k\text{.} \end{gathered}`

Dividing the above equation by -10 we get:

`k=-(\ln ((3)/(4)))/(10)\text{.}`

Therefore, the function that models this behavior is:

`\begin{gathered} m(e)=120e^{-(-\ln ((3)/(4)))\cdot(t)/(10)}=120e^{\ln ((3)/(4))\cdot(t)/(10)}\text{.} \\ \text{for t}\ge0. \end{gathered}`

And its graph is:

Answer: