Since the given equation is
[tex]S=3800e^{-0.05x}[/tex]
S is the amount of the daily sales from ending to x days
Since the form of the exponential function is
[tex]y=ae^x[/tex]
Where a is the initial amount (value y at x = 0)
Then 3800 represents the daily sales when x = 0
Since x = 0 at the ending of the campaign, then
a. The daily sales when the campaign ended is $3800
Since the daily sales will be below half $3800 after x days
Then find half 3800, then equate S by it, then find x
[tex]\begin{gathered} S=\frac{1}{2}(3800) \\ S=1900 \end{gathered}[/tex][tex]1900=3800e^{-0.05x}[/tex]
Divide both sides by 3800
[tex]\begin{gathered} \frac{1900}{3800}=\frac{3800}{3800}e^{-0.05x} \\ \frac{1}{2}=e^{-0.05x} \end{gathered}[/tex]
Insert ln for both sides
[tex]\ln (\frac{1}{2})=\ln (e^{-0.05x})[/tex]
Use the rule
[tex]\ln (e^n)=n[/tex][tex]\ln (\frac{1}{2})=-0.05x[/tex]
Divide both sides by -0.05 to find x
[tex]\begin{gathered} \frac{\ln (\frac{1}{2})}{-0.05}=\frac{-0.05x}{-0.05} \\ 13.86294=x \end{gathered}[/tex]
Since we need it below half 3800, then we round the number up to the nearest whole number
Then x = 14 days
b. 14 days will pass after the campaign ended
