Question

I need help please!!! My teacher doesn't teacher and she has asked us to do a problem I don't know how to do it. Can somebody explain or give the answer and show your work so I can understand that way? I will be giving away 20 points and brainliest. Thank You!
A patient is told to avoid caffeine for 8 to 12 hours before a blood test scheduled for 6 a.m. The blood test is reliable for up to 50 milligrams of caffeine in the bloodstream. The patient’s body metabolizes caffeine at a rate of 13% per hour.
a. At 10 p.m., the patient drinks a cup of coffee containing 150 milligrams of caffeine. Will the patient be ready for the blood test by 6 a.m.? Explain.
b. How many milligrams of caffeine could the patient have ingested at 7 p.m. and been ready for the blood test at 6 a.m.?

Answer 1

Using exponential functions, it is found that:

a) Since the amount of caffeine will be less than 50 mg, the patient will be ready for the blood test by 6 a.m.

b) The patient could have ingest 231 milligrams of caffeine.

A decaying exponential function is modeled by:

$A(t) = A(0)(1 - r)^t$

In which:

• A(0) is the initial value.
• r is the decay rate, as a decimal.

In this problem:

• Caffeine metabolize at a rate of 13% per hour, hence $r = 0.13$.

Then:

$A(t) = A(0)(1 - r)^t$

$A(t) = A(0)(1 - 0.13)^t$

$A(t) = A(0)(0.87)^t$

Item a:

The coffee cup contains 150 milligrams of caffeine, hence $A(0) = 150$.

At 6 a.m., it is 8 hours after drinking the coffee, hence we have to find A(8).

$A(t) = A(0)(0.87)^t$

$A(8) = 150(0.87)^8$

$A(8) = 49.2$

Since the amount of caffeine will be less than 50 mg, the patient will be ready for the blood test by 6 a.m.

Item b:

This A(0), considering A(11) = 50, hence:

$50 = A(0)(0.87)^{11}$

$A(0) = \frac{50}{(0.87)^{11}}$

$A(0) = 231$

The patient could have ingest 231 milligrams of caffeine.

A similar problem is given at brainly.com/question/25537936