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.
1 answer