Question

Ben consumes an energy drink that contains caffeine. After consuming the energy drink, the amount of caffeine in Ben's body decreases exponentially. The 10-hour decay factor for the number of mg of caffeine in Ben's body is 0.2722. What is the 5-hour growth/decay factor for the number of mg of caffeine in Ben's body

Answer 1

The 5-hour decay factor for the number of mg of caffeine in Ben's body is of 0.1469.

Explanation:

After consuming the energy drink, the amount of caffeine in Ben's body decreases exponentially.

This means that the amount of caffeine after t hours is given by:

`A(t) = A(0)e^(-kt)`

In which A(0) is the initial amount and k is the decay rate, as a decimal.

The 10-hour decay factor for the number of mg of caffeine in Ben's body is 0.2722.

1 - 0.2722 = 0.7278, thus,
`A(10) = 0.7278A(0)`. We use this to find k.

`A(t) = A(0)e^(-kt)`

`0.7278A(0) = A(0)e^(-10k)`

`e^(-10k) = 0.7278`

`\ln{e^(-10k)} = ln(0.7278)`

`-10k = ln(0.7278)`

`k = -(ln(0.7278))/(10)`

`k = 0.03177289938`

Then

`A(t) = A(0)e^(-0.03177289938t)`

What is the 5-hour growth/decay factor for the number of mg of caffeine in Ben's body?

We have to find find A(5), as a function of A(0). So

`A(5) = A(0)e^(-0.03177289938*5)`

`A(5) = 0.8531`

The decay factor is:

1 - 0.8531 = 0.1469

The 5-hour decay factor for the number of mg of caffeine in Ben's body is of 0.1469.