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.