Question

Use the graph to estimate how long it would take for your body to only have 50mg of caffeine remaining in your system.Draw this point on your graph from part b.

Answer 1

We have a function that relates the amount of caffeine in the body in function of the time.

This function is:

`A(x)=100e^(-0.17x)`

where A is the amount of caffeine in mg and x is the time in hours.

We have to estimate the time when the amount of caffeine is 50 mg.

This means, in terms of the function, to find x so that A = 50.

We can do it like this:

`\begin{gathered} A(x)=50 \\ 100e^(-0.17x)=50 \\ e^(-0.17x)=\frac{50}{100^{}} \\ e^(-0.17x)=0.5 \\ ln(e^(-0.17x))=\ln (0.5) \\ -0.17x=\ln (0.5) \\ x=-(\ln (0.5))/(0.17) \\ x\approx4.08 \end{gathered}`

The amount of caffeine will be 50 mg after 4.08 hours approximately.

We can graph it as: