Answer:
1) 4 days
2) 20 g
[tex]\textsf{3)}\quad A(t)=320\left(\dfrac{1}{2}\right)^{\dfrac{t}{4}}[/tex]
Step-by-step explanation:
Question 1
The half-life of an element is the time required for the sample to halve.
From the given table, we observe that the initial sample at t = 0 is 320 g. As half of 320 g is 160 g, we need to find the value of t that corresponds to A(t) = 160.
Since A = 160 when t = 4, the half-life of the element is 4 days.
[tex]\hrulefill[/tex]
Question 2
Since the half-life is 4 days, this indicates that the sample halves every 4 days. Consequently, as we add 4 to the t-value, we divide the A-value by 2:
[tex]A(0)=320[/tex]
[tex]A(4)=160[/tex]
[tex]A(8) = 80[/tex]
[tex]A(12)=40[/tex]
[tex]A(16)=20[/tex]
Therefore, after 16 days, there will be 20 g of the element present.
[tex]\hrulefill[/tex]
Question 3
The half-life formula is given by:
[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, the initial quantity is 320 and the half-life is 4 days. Therefore, substitute these values into the formula to create an equation for A(t):
[tex]A(t)=320\left(\dfrac{1}{2}\right)^{\dfrac{t}{4}}[/tex]
