A drug is eliminated from the body through the kidney in such a way that over each hour, 25% of the amount present at the beginning of the hour is eliminated. Let t be the amount of time (in hours) since the drug was first taken, and A0 the initial amount of the drug in the body.

Question

A drug is eliminated from the body through the kidney in such a way that over each hour, 25% of the amount present at the beginning of the hour is eliminated. Let t be the amount of time (in hours) since the drug was first taken, and A0 the initial amount of the drug in the body.

A(t) = A0e^(kt)

k is some constant. How long does it take before the amount of drug in the body is half of the initial amount?

Answer 1

A(t) = A0 (0.75)^t = A0 (e^ln(0.75))^t = A0 e^(ln(0.7t) t)

so, k = ln(0.75)

now you can find when A(t) = 0.5 A0

Answer 2

To find out how long it takes before the amount of drug in the body is half of the initial amount, we need to solve for t in the equation A(t) = 0.5A0.

A(t) = A0e^(kt)

Replacing A(t) with 0.5A0, we get:

0.5A0 = A0e^(kt)

To simplify the equation, divide both sides by A0:

0.5 = e^(kt)

To isolate the exponent e^(kt), take the natural logarithm (ln) of both sides:

ln(0.5) = ln(e^(kt))

Apply the logarithmic property, ln(e^(kt)) = kt:

ln(0.5) = kt

Now, divide both sides by k:

t = ln(0.5) / k

Therefore, it takes t = ln(0.5) / k hours for the amount of drug in the body to be half of the initial amount.

Answer 3

To find the time it takes for the amount of drug in the body to be half of the initial amount, we can use the equation A(t) = A0e^(kt) given in the problem statement.

Since we're looking for the time when A(t) is half of A0, we have:

A(t) = (1/2)A0

Substituting this into the equation, we get:

(1/2)A0 = A0e^(kt)

Next, we can simplify the equation by canceling out A0 on both sides:

1/2 = e^(kt)

To solve for t, we need to isolate the exponential term. We can do this by taking the natural logarithm (ln) of both sides:

ln(1/2) = ln(e^(kt))

Using the property of logarithms, ln(e^(kt)) simplifies to kt * ln(e):

ln(1/2) = kt * 1

Now, divide both sides of the equation by k to solve for t:

t = ln(1/2) / k

Finally, we have the value of t in terms of k. Since k is a constant specific to the drug, we would need additional information or data about the drug to determine its value and calculate the exact time it takes for the drug to be eliminated by half.