4) A summer camp rewards campers and zounselors with badges. The carmp orders 200 badges. They plan to give 25 badges to counselors. They ordered at least 3 badges for each camper. a. How many campers could be at the camp? Show your work. SOLUTION $$ \qquad $$ b. Are all of the values on the graph of $$ c \leq 58 $$ possible solutions? Explain.

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To solve the problem, let's break it down step by step.

### Part a: How many campers could be at the camp?

1. **Identify the known conditions:**
   - Total badges ordered: $$ 200 $$
   - Badges given to counselors: $$ 25 $$
   - Badges for each camper: at least $$ 3 $$

2. **Calculate the number of badges available for campers:**
   $$
   \text{Badges for campers} = \text{Total badges} - \text{Badges for counselors} = 200 - 25 = 175
   $$

3. **Let $$ c $$ be the number of campers. Since each camper receives at least $$ 3 $$ badges, we can express this as:**
   $$
   3c \leq 175
   $$

4. **Now, solve for $$ c $$:**
   $$
   c \leq \frac{175}{3}
   $$
   $$
   c \leq 58.33
   $$

5. **Since the number of campers must be a whole number, the maximum number of campers is:**
   $$
   c \leq 58
   $$

Thus, the maximum number of campers that could be at the camp is $$ 58 $$.

### Part b: Are all of the values on the graph of $$ c \leq 58 $$ possible solutions? Explain.

The inequality $$ c \leq 58 $$ indicates that any whole number value of $$ c $$ from $$ 0 $$ to $$ 58 $$ is a possible solution.

1. **Possible values for $$ c $$:** 
   - The values can be $$ 0, 1, 2, \ldots, 58 $$.

2. **Explanation:**
   - Each of these values represents a scenario where the campers receive at least $$ 3 $$ badges each, and the total number of badges does not exceed $$ 200 $$.
   - For example, if $$ c = 58 $$, then the total badges used would be $$ 25 + 3 \times 58 = 25 + 174 = 199 $$, which is within the limit.
   - If $$ c = 0 $$, then all badges go to counselors, which is also valid.

Therefore, all values on the graph of $$ c \leq 58 $$ are indeed possible solutions, as they all satisfy the condition of not exceeding the total number of badges available.
a. The camp could have up to 58 campers. b. Yes, all values of $$ c \leq 58 $$ are possible solutions because they meet the badge distribution requirements.

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