A particular pain relieving medicine has a decay rate of 12% per hour. A patient was given a dose of the medicine 5 hours ago and there is currently 80 milligrams of the medicine in the patients bloodstream.

After the dose was given how long must the patient wait for there to be less than 25% of the original dose of the medicine left in his or her bloodstream? Express your answer as a decimal to the nearest tenth of an hour.

amount at t hours = original amount * 0.88^t

so
fraction at time t = 0.88^t

if

0.25 = 0.88^t
then
log 0.25 = t log 0.88

(1 - .12)^t = .25 ... t [log(1 - .12)] = log(.25) ... t = log(.88) / log(.25)

To solve this problem, we need to find the time it takes for the medicine to decay to less than 25% of the original dose.

Let's start by calculating the current amount of medicine in the bloodstream, given that it decays at a rate of 12% per hour for 5 hours.

After 5 hours, the remaining amount of medicine can be calculated using the formula: Remaining amount = Initial amount * (1 - Decay rate)^Time

Remaining amount = 80 * (1 - 0.12)^5
Remaining amount = 80 * (0.88)^5
Remaining amount ≈ 80 * 0.42
Remaining amount ≈ 33.6 milligrams

Now, we want to find the time it takes for the medicine to decay to less than 25% of the original dose, which is 25% of 80 milligrams = 20 milligrams.

Using the same formula, we can solve for the time:

Remaining amount = Initial amount * (1 - Decay rate)^Time
20 = 80 * (1 - 0.12)^Time

Next, divide both sides of the equation by 80:

20/80 = (1 - 0.12)^Time

Simplify:

0.25 = 0.88^Time

To solve for Time, we need to take the logarithm of both sides:

log(0.25) = log(0.88^Time)

Using the property of logarithms, we can bring down the exponent:

log(0.25) = Time * log(0.88)

Now, divide both sides by log(0.88):

Time = log(0.25) / log(0.88)

Using a calculator, we can find:

Time ≈ -0.602 / -0.056

Time ≈ 10.75

So, the patient must wait for approximately 10.8 hours (rounded to the nearest tenth of an hour) for there to be less than 25% of the original dose of the medicine left in their bloodstream.