Question

a doctor prescribes 175 milligrams of a therapeutic drug that decays by 25% each hour. what is the half-life of the drug? round to the nearest hundredth. hours what is the amount of therapeutic drug left after 10 hours? round to the nearest hundredth.

Explanation:

To solve this, we first need to know the initial dosage (let's assume it's missing in the provided text, but let's assume the initial dosage is \( N_0 \) milligrams, and the decay rate \( r = 25\%= 0.25 \) per hour. The formula for exponential decay is \( N(t)=N_0(1 - r)^t \), where \( t \) is time in hours. Also, the half - life formula for exponential decay is \( T_{1/2}=\frac{\ln(0.5)}{\ln(1 - r)} \)

Step 1: Find the half - life

We know that the decay rate \( r = 0.25 \), so \( 1-r=1 - 0.25 = 0.75\)

The formula for half - life \( T_{1/2}=\frac{\ln(0.5)}{\ln(0.75)} \)

We know that \( \ln(0.5)\approx - 0.6931\) and \( \ln(0.75)\approx - 0.2877\)

\( T_{1/2}=\frac{- 0.6931}{- 0.2877}\approx2.41 \) hours

Step 2: Find the amount left after 10 hours

Let's assume the initial dosage \( N_0\) (since the problem statement seems to have a typo and the initial dosage is not provided. Let's assume the initial dosage \( N_0 = 100\) mg (we can work with any initial value, the ratio will be the same). The formula for the amount of drug left after \( t\) hours is \( N(t)=N_0(1 - r)^t\)

\( r = 0.25\), \( t = 10\)

\( N(10)=N_0\times(0.75)^{10}\)

\( (0.75)^{10}\approx0.0563\)

If \( N_0 = 100\) mg, then \( N(10)=100\times0.0563 = 5.63\) mg

Since the initial problem statement has a typo (the initial number of milligrams is missing), if we assume the initial dosage \( N_0\) (for example, if the initial dosage was \( N_0 = 100\) mg):

Answer:

• Half - life: \(\approx\boldsymbol{2.41}\) hours
• Amount left after 10 hours (assuming \( N_0 = 100\) mg): \(\approx\boldsymbol{5.63}\) mg

If you can provide the initial number of milligrams of the drug, we can give a more accurate answer for the amount left after 10 hours.