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:

Question

$$A=A_0\times (1-r{)}^n$$

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:

$$\boxed{{\displaystyle 63.28}}\text{ mg}$$
make this like a 7th grader wrote it

Answer 1

To find out how much antibiotic is left in the dog after 16 hours, we can use a simple formula that helps us understand how the amount decreases over time. The formula is:

$$A=A_0\times (1-r{)}^n$$

Here's what the parts mean:

• $A$ is how much antibiotic is left.
• $A_0$ is how much antibiotic we started with.
• $r$ is how quickly it decays (like how much is lost).
• $n$ is how many time intervals we count.

Now, let's use the numbers given:

• The starting amount $A_0$ is 200 mg.
• The decay rate $r$ is 0.25, which means 25% of the antibiotic is gone each time period.
• The total time $t$ we are looking at is 16 hours.

First, we need to figure out how many time intervals of 4 hours fit into 16 hours:

$$n=\frac{t}{4}=\frac{16}{4}=4$$

This means there are 4 intervals of 4 hours in 16 hours.

Next, we can put the values into the equation:

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

First, let's calculate $(1-0.25)$:

$$1-0.25=0.75$$

Now, we plug that back into our equation:

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

Now we need to calculate $(0.75{)}^4$:

$$(0.75{)}^4=0.75\times 0.75\times 0.75\times 0.75$$

Doing the math step by step:

1. $0.75\times 0.75=0.5625$
2. $0.5625\times 0.75=0.421875$
3. $0.421875\times 0.75=0.31640625$

Now we can substitute this back into our equation:

$$A=200\times 0.31640625$$

Calculating that gives us:

$$A\approx 63.28125$$

So, the amount of antibiotic left in the dog after 16 hours is about:

$$\boxed{{\displaystyle 63.28}}\text{ mg}$$

That's how much is still in the dog after all that time!