Answer:
[tex]f^{-1}(x)=3\frac{ln(\frac{t}{80})}{ln(2)}[/tex]
It will take 20.9 hours to reach 10000
Step-by-step explanation:
[tex]f(t)=80(2)^{\frac{t}{3}}[/tex]
To find inverse of the function , replace f(x) with y
[tex]y=80(2)^{\frac{t}{3}}[/tex]
swap the variables
[tex]t=80(2)^{\frac{y}{3}}[/tex]
solve the equation for y
[tex]t=80(2)^{\frac{y}{3}}\\\\\frac{t}{80} =(2)^{\frac{y}{3}}[/tex]
take ln on both sides
[tex]ln(\frac{t}{80} )= \frac{y}{3} ln(2)\\\\\frac{ln(\frac{t}{80} }{ln(2)} =\frac{y}{3}\\3\frac{ln(\frac{t}{80} }{ln(2)} =y[/tex]
the inverse function is
[tex]f^{-1}(x)=3\frac{ln(\frac{t}{80})}{ln(2)}[/tex]
for part b plug in 10000 for t in f^-1(x)
[tex]f^{-1}(10000)=3\frac{ln(\frac{t}{80} }{ln(2)} \\f^{-1}(10000)=3\frac{ln(\frac{10000}{80} }{ln(2)} \\=20.89[/tex]
It will take 20.9 hours to reach 10000
