A Particular pain relieving medicine has a decay rate of 24% an hour. A patient was given a dose of the medicine 5 hours ago and there is currently 74mg of the medicine in the patients bloodstream.

Q: What was the Origional dose of the medicine? ________ milligrams.
Q How long will the patient need to wait for there to be less than 25% of the original dose of the medicine left in the bloodstream? _______hours. (to the nearest tenth of an hour)

after t hours, starting with A mg, the amount remaining is

A*.76^t

So, to find the original dose, solve

A*.76^5 = 74

To get to 25%, you want

.76^t = .25

To find the original dose of the medicine, we can use the concept of exponential decay.

Let's start with the formula for exponential decay:

Amount = Initial Amount * e^(decay rate * time)

where:
- Amount is the current amount of the medicine in the bloodstream,
- Initial Amount is the original dose of the medicine,
- e is the mathematical constant approximately equal to 2.71828,
- Decay rate is the rate at which the medicine decays per unit of time,
- Time is the elapsed time since the medicine was administered.

For the first question, we are given that the current amount of the medicine in the bloodstream is 74mg after 5 hours and the decay rate is 24% per hour.

Using the exponential decay formula, we can solve for the initial dose:

74mg = Initial Amount * e^(0.24 * 5)

To solve this equation, we can divide both sides by e^(0.24 * 5):

Initial Amount = 74mg / e^(0.24 * 5)

Using a calculator, we can find the value of e^(0.24 * 5) ≈ 1.6355.

So, the original dose of the medicine is:

Initial Amount = 74mg / 1.6355 ≈ 45.21mg (rounded to two decimal places)

Therefore, the original dose of the medicine was approximately 45.21 milligrams.

For the second question, we need to find how long the patient needs to wait until there is less than 25% of the original dose left in the bloodstream.

Using the exponential decay formula, we can rearrange it to solve for time:

Time = (ln(Amount) - ln(Desired Amount)) / (decay rate)

where:
- Desired Amount is the 25% of the original dose.

Substituting the values into the equation:

Time = (ln(74mg) - ln(0.25 * Initial Amount)) / (0.24)

Using a calculator, we can find the natural logarithm of 74mg ≈ 4.3041.

So, the time required for there to be less than 25% of the original dose left in the bloodstream is:

Time = (4.3041 - ln(0.25 * Initial Amount)) / 0.24

Substituting the value of the initial dose we found earlier, we get:

Time = (4.3041 - ln(0.25 * 45.21mg)) / 0.24

Using a calculator, we find ln(0.25 * 45.21mg) ≈ -2.7627.

Time = (4.3041 - (-2.7627)) / 0.24 ≈ 28.2329 hours

Rounded to the nearest tenth of an hour, the patient needs to wait approximately 28.2 hours until there is less than 25% of the original dose left in the bloodstream.