A drug is eliminated from the body through urine. Suppose that for a dose of 10 milligrams, the amount A(t) remaining in the body t hours later is given by
A(t) = 10(0.7)^t and that in order for the drug to be effective, at least 2 milligrams must be in the body.
ok. Now what?
I assume you want to know how long it will take to drop to the 2mg level.
So, just solve
10 * 0.7^t = 2
0.7^t = 0.2
now just take the last step.
To find out how long it takes for the drug to be eliminated from the body, we need to find the time when the amount remaining (A(t)) is less than 2 milligrams.
Given that A(t) = 10(0.7)^t, we can substitute 2 milligrams for A(t) and solve for t:
2 = 10(0.7)^t
Now we can solve for t using logarithms:
Divide both sides by 10:
2/10 = (0.7)^t
Rearrange the equation:
0.2 = (0.7)^t
Take the logarithm of both sides (preferably logarithm with base 10):
log(0.2) = log((0.7)^t)
Using logarithmic properties, we can bring down the exponent:
log(0.2) = t * log(0.7)
Now solve for t by dividing both sides by log(0.7):
t = log(0.2) / log(0.7)
Using a scientific calculator or a calculator software that supports logarithmic functions, you can evaluate this expression:
t ≈ -2.51 hours
Since time cannot be negative, we take the absolute value to get the positive value:
t ≈ 2.51 hours
Therefore, it takes approximately 2.51 hours for the drug to be eliminated from the body.