QuestionPart B: A certain medication comes in 200 mg tablets. This drug has a half-life of three years. You decide to throw the drug out once its strength decays to 150 milligrams. How long will it take to reach that level? Give your answer in years and months.
Answer: $\square$ 1 os year(s) and $\square$ months Enter an integer or decimal number [more..] Question Help: Message instructor Calculator Submit Question

Studdy Solution

STEP 1

1. The medication comes in 200 mg tablets.
2. The drug has a half-life of three years.
3. The drug is thrown out once its strength decays to 150 mg.
4. We need to find the time it takes for the drug to decay from 200 mg to 150 mg.
5. The answer should be provided in years and months.

STEP 2

1. Understand the concept of half-life and its application.
2. Use the half-life formula to set up an equation.
3. Solve the equation for time.
4. Convert the time from years to years and months.

STEP 3

Understand the concept of half-life. The half-life of a substance is the time it takes for half of the substance to decay. In this problem, the half-life is given as three years.

STEP 4

Use the half-life formula to set up an equation. The formula for exponential decay is:
$A = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}$
where: - $A$ is the final amount of the substance. - $A_0$ is the initial amount of the substance. - $t$ is the time elapsed. - $T$ is the half-life of the substance.
Given: - $A = 150$ mg - $A_0 = 200$ mg - $T = 3$ years
Substitute these values into the formula:
$150 = 200 \left(\frac{1}{2}\right)^{\frac{t}{3}}$

STEP 5

Solve the equation for time $t$ .
First, divide both sides by 200:
$\frac{150}{200} = \left(\frac{1}{2}\right)^{\frac{t}{3}}$
Simplify the left side:
$0.75 = \left(\frac{1}{2}\right)^{\frac{t}{3}}$
Take the natural logarithm of both sides:
$\ln(0.75) = \ln\left(\left(\frac{1}{2}\right)^{\frac{t}{3}}\right)$
Use the property of logarithms to bring down the exponent:
$\ln(0.75) = \frac{t}{3} \ln\left(\frac{1}{2}\right)$
Solve for $t$ :
$t = 3 \frac{\ln(0.75)}{\ln\left(\frac{1}{2}\right)}$
Calculate $t$ :
$t \approx 1.7095 \text{ years}$

STEP 6

Convert the time from years to years and months.
The integer part of $t$ is 1 year. To find the months, take the decimal part and multiply by 12:
$0.7095 \times 12 \approx 8.5 \text{ months}$
Round to the nearest whole number:
$\approx 9 \text{ months}$
The time it takes for the drug to decay to 150 mg is:
$\boxed{1}$ year(s) and $\boxed{9}$ months

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