The function $$ f(x)=500(0.92)^{x} $$ gives the number of milligrams of the active ingredient in a medication in a dog's consumed a chewable tablet. Which statements about this situation are true? Select all that apply. $$ \square $$ There are 500 milligrams of the active ingredient in the dog's body immediately after it consumes the tablet. $$ \square $$ The amount of active ingredient in the dog's body changes by 500 milligrams per hour. $$ \square $$ The amount of active ingredient decreases by 0.92 milligrams per hour. The amount of active ingredient decreases by $$ 8 \% $$ per hour. The amount of active ingredient decreases by $$ 92 \% $$ per hour.

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Let's analyze each statement based on the function $$ f(x) = 500(0.92)^x $$, where $$ x $$ represents the number of hours:

1. **There are 500 milligrams of the active ingredient in the dog's body immediately after it consumes the tablet.**
   
   - **True.** When $$ x = 0 $$:
     $$
     f(0) = 500(0.92)^0 = 500 \times 1 = 500 \, \text{milligrams}
     $$
     This represents the initial amount of the active ingredient.

2. **The amount of active ingredient in the dog's body changes by 500 milligrams per hour.**
   
   - **False.** The function describes an exponential decay, not a linear change. The amount does not decrease by a fixed number of milligrams each hour.

3. **The amount of active ingredient decreases by 0.92 milligrams per hour.**
   
   - **False.** The decay is multiplicative, not additive. The amount decreases by a certain percentage each hour, not a fixed milligram amount.

4. **The amount of active ingredient decreases by $$ 8\% $$ per hour.**
   
   - **True.** Each hour, the amount is multiplied by 0.92, meaning it retains 92% of the previous hour's amount. Therefore, it decreases by:
     $$
     100\% - 92\% = 8\% \text{ per hour}
     $$

5. **The amount of active ingredient decreases by $$ 92\% $$ per hour.**
   
   - **False.** As established, the amount retains 92% each hour, meaning it only decreases by 8%.

**Correct Statements:**
- There are 500 milligrams of the active ingredient in the dog's body immediately after it consumes the tablet.
- The amount of active ingredient decreases by $$ 8\% $$ per hour.

**Final Answer:**
Only the first and the fourth statements are true.
The first and fourth statements are true: - There are 500 milligrams of the active ingredient immediately after consumption. - The amount decreases by 8% each hour.

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