1)A particular pain relieving medicine has a decay rate of 10 % per hour. A patient was given a dose of the medicine 7 hours ago and there is currently 68 milligrams of the medicine in the patients bloodstream.
2)How long will the patient need to wait for there to be less than 25 % of the original dose of the medicine left in their bloodstream? Express your answer as a decimal to the nearest tenth of an hour.
.25 > (1 - .10)^(t + 7)
log(.25) > (t + 7) log(.9)
To solve this problem, we need to use the concept of exponential decay. The equation we can use to represent the amount of medicine remaining in the patient's bloodstream after a certain time is:
M(t) = M0 * (1 - r)^t
where M(t) is the amount of medicine remaining at time t, M0 is the initial amount of medicine, r is the decay rate per hour as a decimal (in this case, 10% = 0.10), and t is the time in hours.
1) We know that the patient was given a dose of the medicine 7 hours ago, and there is currently 68 milligrams of the medicine in the patient's bloodstream. Plugging these values into the equation, we have:
68 = M0 * (1 - 0.10)^7
To find M0, we can rearrange the equation:
M0 = 68 / (1 - 0.10)^7
Calculating this, we get M0 ≈ 202.74 milligrams as the initial dose of the medicine.
2) Now, we want to find out how long it will take for there to be less than 25% of the original dose left in the patient's bloodstream. In other words, we want to find t when M(t) is less than 25% of M0.
25% of M0 is 0.25 * 202.74 ≈ 50.69 milligrams.
Plugging this value into the equation, we have:
50.69 = 202.74 * (1 - 0.10)^t
To solve for t, we need to isolate t on one side of the equation. First, divide both sides of the equation by 202.74:
(1 - 0.10)^t = 50.69 / 202.74
Next, take the natural logarithm (ln) of both sides to remove the exponent:
ln((1 - 0.10)^t) = ln(50.69 / 202.74)
Using the property of logarithm, we can bring down the t:
t * ln(1 - 0.10) = ln(50.69 / 202.74)
Divide both sides by ln(1 - 0.10):
t = ln(50.69 / 202.74) / ln(1 - 0.10)
Calculating this, we find t ≈ 5.7 hours. Therefore, the patient will need to wait for approximately 5.7 hours for there to be less than 25% of the original dose of the medicine left in their bloodstream.