Polonium-210 has a half-life of 140 days. Suppose a sample of this substance has a mass of 300 mg-

1.Find a function that models the amount of the sample remaining at time t.

2.Find the mass remaining after 1 year.

3.How long will it take for the sample to decay to a mass of 200 mg?

Respuesta :

Answer:

[tex]\textsf{1)}\quad A(t)=300\left(\dfrac{1}{2}\right)^{\dfrac{t}{140}}\qquad \textsf{(where $t$ is the number of days)}[/tex]

2)  49.2 g (3 s.f.)

3)  81.9 days (3 s.f.)

Step-by-step explanation:

Question 1

To find a function that models the amount of the sample remaining at time t, we can use the half-life formula:

[tex]\boxed{\begin{array}{l}\underline{\textsf{Half Life Formula}}\\\\A(t)=A_0\left(\dfrac{1}{2}\right)^{\dfrac{t}{t_{\frac12}}}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$A(t)$ is the quantity remaining.}\\\phantom{ww}\bullet\;\textsf{$A_0$ is the initial quantity.}\\ \phantom{ww}\bullet\;\textsf{$t$ is the time elapsed.}\\\phantom{ww}\bullet\;\textsf{$t_{\frac12}$ is the half-life of the substance.}\end{array}}[/tex]

In this case:

[tex]A_0 = 300 \;\sf mg[/tex]

[tex]t_{\frac12}=140\; \sf days[/tex]

Substitute these values into the formula:

[tex]A(t)=300\left(\dfrac{1}{2}\right)^{\dfrac{t}{140}}[/tex]

[tex]\hrulefill[/tex]

Question 2

To find the mass remaining after 1 year, we can substitute t = 365 into the function A(t), since there are 365 days in 1 year:

[tex]A(365)=300\left(\dfrac{1}{2}\right)^{\dfrac{365}{140}}\\\\\\A(365)=49.237166447...\\\\\\A(365)=49.2\; \sf g\;(3\;s.f.)[/tex]

Therefore, the mass remaining after 1 year is 49.2 g (rounded to three significant figures).

[tex]\hrulefill[/tex]

Question 3

To find how long will it take for the sample to decay to a mass of 200 mg, substitute A(t) = 200 into the formula and solve for t:

[tex]300\left(\dfrac{1}{2}\right)^{\dfrac{t}{140}}=200\\\\\\\\\left(\dfrac{1}{2}\right)^{\dfrac{t}{140}}=\dfrac{2}{3}\\\\\\\\\ln\left(\left(\dfrac{1}{2}\right)^{\dfrac{t}{140}\right)=\ln\left(\dfrac{2}{3}\right)\\\\\\\\\dfrac{t}{140}\ln\left(\dfrac{1}{2}\right)=\ln\left(\dfrac{2}{3}\right)\\\\\\\\t=\dfrac{140\ln\left(\frac{2}{3}\right)}{\ln\left(\frac{1}{2}\right)}\\\\\\\\t=81.89475010...\\\\\\t=81.9\; \sf days\;(3\;s.f.)[/tex]

Therefore, it will take approximately 81.9 days for the sample to decay to a mass of 200 mg.