To determine the amount of antibiotics left in the dog after 16 hours, we can use the formula for exponential decay. The formula for the amount remaining after time \( t \) is:
\[ A = A_0 \times (1 - r)^n \]
where:
- \( A \) is the amount remaining,
- \( A_0 \) is the initial amount,
- \( r \) is the decay rate,
- \( n \) is the number of time intervals.
Given:
- \( A_0 = 200 \) mg (initial amount),
- \( r = 0.25 \) (25% decay rate),
- \( t = 16 \) hours.
First, we need to determine \( n \), which is the number of 4-hour intervals in 16 hours:
\[ n = \frac{t}{4} = \frac{16}{4} = 4 \]
Now we can substitute the values into the formula:
\[ A = 200 \times (1 - 0.25)^4 \]
Calculating \( (1 - 0.25) \):
\[ 1 - 0.25 = 0.75 \]
Now substituting back into the formula:
\[ A = 200 \times (0.75)^4 \]
Calculating \( (0.75)^4 \):
\[ (0.75)^4 = 0.75 \times 0.75 \times 0.75 \times 0.75 = 0.31640625 \]
Now substitute this value into the equation:
\[ A = 200 \times 0.31640625 \approx 63.28125 \]
Thus, the amount of antibiotic left in the dog after 16 hours is approximately:
\[ \boxed{63.28} \text{ mg} \]