Questions: A veterinarian has to give antibiotics to a dog with a leg infection. The dog is given 200 mg, and the antibiotic decays at a rate of 25 percent every 4 hours. What amount of medicine is left in the dog after 16 hours? (1 point) - 63.3 mg - 488.28 mg - 0.78 mg - 2 mg

Solution Steps

The problem involves exponential decay of a substance. The initial amount of the antibiotic is 200 mg, and it decays at a rate of 25% every 4 hours. We need to find the amount of medicine left after 16 hours.

The formula for exponential decay is given by:

\[ A = A_0 \left(1 - \frac{r}{100}\right)^t \]

where:

• \( A \) is the amount remaining after time \( t \),
• \( A_0 \) is the initial amount,
• \( r \) is the decay rate (in percentage),
• \( t \) is the number of decay periods.

Since the decay rate is given for every 4 hours, and we need to find the amount after 16 hours, we calculate the number of decay periods:

\[ t = \frac{16 \text{ hours}}{4 \text{ hours/period}} = 4 \text{ periods} \]

Substitute the known values into the decay formula:

\[ A = 200 \left(1 - \frac{25}{100}\right)^4 \]

Simplify the expression:

\[ A = 200 \times (0.75)^4 \]

Calculate \( (0.75)^4 \):

\[ (0.75)^4 = 0.3164 \]

Now, calculate \( A \):

\[ A = 200 \times 0.3164 = 63.28 \text{ mg} \]

Final Answer