Question

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)
Responses

0.78 mg
0.78 mg

2 mg
2 mg

488.28 mg
488.28 mg

63.3 mg

Answer 1

To determine how much antibiotic is left in the dog after 16 hours, given that it decays at a rate of 25 percent every 4 hours, we can use the formula for exponential decay.

The amount of antibiotic left after \( t \) hours can be calculated as follows:

\[
A = A_0 \times (1 - r)^{(t/T)}
\]

Where:
- \( A_0 \) is the initial amount (200 mg),
- \( r \) is the decay rate (0.25 for 25%),
- \( T \) is the time interval for the decay (4 hours),
- \( t \) is the total time (16 hours).

First, we need to determine how many 4-hour intervals are in 16 hours:

\[
\frac{t}{T} = \frac{16 \text{ hours}}{4 \text{ hours}} = 4
\]

Now substituting the known values into the formula:

\[
A = 200 \times (1 - 0.25)^{4}
\]

Calculating \( (1 - 0.25) \):

\[
1 - 0.25 = 0.75
\]

Now substitute back into the equation:

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

Calculating \( (0.75)^{4} \):

\[
(0.75)^{4} = 0.31640625
\]

Now, multiply:

\[
A = 200 \times 0.31640625 \approx 63.28125
\]

Rounding this to two decimal places, the amount left is approximately \( 63.28 \) mg.

Among the provided options, the closest value is:

**63.3 mg**.