2 hours after a given drug has been administered there is only 75% of it left in the body. a. What percentage of it leaves the body every hour (find the hourly decay rate)?
b. What is the continuous decay rate?
c. How long until 50% has left the body?
d. How long until 99% of it has left the body?
a. To find the hourly decay rate, we need to determine how much of the drug is leaving the body per hour. We know that after 2 hours, only 75% of the drug is left. This means that in 2 hours, 25% of the drug has left the body (100% - 75% = 25%). Therefore, we can divide the percentage of drug leaving the body (25%) by the number of hours (2) to find the hourly decay rate.
Hourly Decay Rate = Percentage of Drug Leaving the Body / Number of Hours
Hourly Decay Rate = 25% / 2 = 12.5%
So, the percentage of the drug leaving the body every hour is 12.5%.
b. The continuous decay rate refers to the rate at which the drug is continually leaving the body over time. Since we already know the hourly decay rate to be 12.5%, we can convert it to a continuous decay rate.
Continuous Decay Rate = Hourly Decay Rate / 100
Continuous Decay Rate = 12.5% / 100 = 0.125
So, the continuous decay rate is 0.125.
c. To find out how long until 50% of the drug has left the body, we need to determine the number of hours it takes for the remaining drug to reach 50%. Since we know the hourly decay rate is 12.5%, we can set up an equation to solve for the time.
Remaining Percentage = Initial Percentage * (1 - Hourly Decay Rate)^Time
Substituting the given values:
50% = 100% * (1 - 12.5%)^Time
Simplifying:
0.5 = 0.875^Time
Using logarithms, we can solve for Time:
Time = log(0.5) / log(0.875)
By evaluating this expression, we find that Time ≈ 4.545.
So, it takes approximately 4.545 hours for 50% of the drug to leave the body.
d. To find out how long until 99% of the drug has left the body, we can use a similar approach as in part c.
Remaining Percentage = Initial Percentage * (1 - Hourly Decay Rate)^Time
Substituting the given values:
1% = 100% * (1 - 12.5%)^Time
Simplifying:
0.01 = 0.875^Time
Using logarithms, we can solve for Time:
Time = log(0.01) / log(0.875)
By evaluating this expression, we find that Time ≈ 33.693.
So, it takes approximately 33.693 hours for 99% of the drug to leave the body.