Question

A cup of coffee contains 130 mg of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how much caffeine will remain in the body after 3 hours?

Write the exponential function and solve.

Answer 1

OK, so first: if 11% of caffeine is eliminated, then 89% stays. So for the first hour,\[130 \times \left(\dfrac{89}{100}\right)\]is there in the coffee. Now, again, 89% of THAT will stay after the next hour. So in 2 hours,\[130 \times \left(\dfrac{89}{100}\right) \times \left(\dfrac{89}{100} \right)\]then in 3 hours, 89% of THAT stays... so\[130 \times \left(\dfrac{89}{100}\right) \times \left(\dfrac{89}{100}\right) \times \left(\dfrac{89}{100}\right)\]is there. Do you see a pattern here?

Answer 2

In \(x \) hours, we see that\[130 \times \underbrace{\left(\dfrac{89}{100}\right) \times \left(\dfrac{89}{100}\right) \times \cdots \left(\dfrac{89}{100}\right) \times \left(\dfrac{89}{100}\right)}_{\Large x- \rm times}\]is there.

Answer 3

Or,\[130\times \left(\dfrac{89}{100}\right)^{\large x}\]is there in \(x\) hours.

Answer 4

Plug in 3 for \(x\) in the above expression, which returns the amount of caffeine in \(x\) hours because we need the amount of caffeine in 3 hours.

Answer 5

so i just do 130x(89/100)^x in my caculator?

Answer 6

3 instead of \(x\)

Answer 7

okay so 130 x (89/100)^3

Answer 8

exactly. I was telling you how this works.

Answer 9

i got 91!

Answer 10

thanks

Answer 11

i guess i round it up to 92

Answer 12

Right, around 91.

Answer 13

Good job.