https://file.aiquickdraw.com/mathbot/file/1726002230898c7qj2mm0.jpg

What does Dogs and Dudes charge for a 3 hour stay?  A 6 hour stay?
What does Puppy Love charge for a 3 hour stay?  A 6 hour stay?
What does Hundreds of Hounds charge for a 3 hour stay?  A 6 hour stay?
Consider the Dogs and Dudes doggy daycare.  Is the amount charged proportional to the number of hours at the daycare?  How can you tell?
Consider the Puppy Love doggy daycare.  Is the amount charged proportional to the number of hours at the daycare?  How can you tell?
Consider the Hundreds of Hounds doggy daycare.  Is the amount charged proportional to the number of hours at the daycare?  How can you tell?
What are the tell-tale signs that two variables are proportional?  List as many as you can!

Let's break down the charges for each daycare using the graph.

### 1. **Dogs and Dudes (Red Line)**
To find the cost for Dogs and Dudes, observe where the red line intersects the vertical axis (y-axis). The slope of the line determines the rate per hour.

- **At 3 hours:** From the graph, the cost for a 3-hour stay at Dogs and Dudes is approximately **$35**.
- **At 6 hours:** The cost for a 6-hour stay at Dogs and Dudes is approximately **$55**.

### 2. **Puppy Love (Blue Line)**
For Puppy Love, we do the same by analyzing the blue line.

- **At 3 hours:** The cost for a 3-hour stay at Puppy Love is approximately **$25**.
- **At 6 hours:** The cost for a 6-hour stay at Puppy Love is approximately **$45**.

### 3. **Hundreds of Hounds (Green Line)**
Now for Hundreds of Hounds, we check the green line.

- **At 3 hours:** The cost for a 3-hour stay at Hundreds of Hounds is approximately **$20**.
- **At 6 hours:** The cost for a 6-hour stay at Hundreds of Hounds is approximately **$35**.

---

### 4. **Proportionality: Dogs and Dudes**
- **Is the charge proportional to the hours?** Yes. The graph is a straight line passing through the origin (0,0), indicating the charge increases at a constant rate per hour (i.e., slope remains constant). This is a sign of proportionality.

### 5. **Proportionality: Puppy Love**
- **Is the charge proportional to the hours?** No. The blue line does not pass through the origin, so the relationship between hours and cost is not proportional. There is a fixed cost or base charge before the hourly rate kicks in.

### 6. **Proportionality: Hundreds of Hounds**
- **Is the charge proportional to the hours?** No. Like Puppy Love, the green line does not pass through the origin, indicating a fixed cost along with the hourly rate.

---

### **Signs that Two Variables Are Proportional:**
1. **Constant Slope (Rate of Change):** The graph is a straight line that passes through the origin (0,0).
2. **Direct Variation:** For every increase in one variable, there is a proportional and consistent increase in the other.
3. **Equation Form:** The equation of the relationship is of the form $ y = kx $, where $ k $ is the constant of proportionality.
4. **No Y-Intercept (Other than Zero):** If there's no starting fixed cost and only a rate based on time, the relationship is proportional.

Would you like to explore any of these details further? 

Here are five related questions you could consider:
1. What is the equation of the line for Dogs and Dudes?
2. How does the slope of each line relate to the hourly cost of each daycare?
3. What would be the cost for an 8-hour stay at Puppy Love?
4. How much more expensive is Dogs and Dudes compared to Hundreds of Hounds for a 5-hour stay?
5. Can you find the proportional constant (slope) for each daycare?

**Tip:** Proportional relationships always have lines passing through the origin on a graph!

What does Dogs and Dudes charge for a 3 hour stay? A 6 hour stay? What does Puppy Love charge for a 3 hour stay? A 6 hour stay? What does Hundreds of Hounds charge for a 3 hour stay? A 6 hour stay? Consider the Dogs and Dudes doggy daycare. Is the amount charged proportional to the number of hours at the daycare? How can you tell? Consider the Puppy Love doggy daycare. Is the amount charged proportional to the number of hours at the daycare? How can you tell? Consider the Hundreds of Hounds doggy daycare. Is the amount charged proportional to the number of hours at the daycare? How can you tell? What are the tell-tale signs that two variables are proportional? List as many as you can!

Explore how to calculate the cost of different stays at doggy daycares (Dogs and Dudes, Puppy Love, Hundreds of Hounds) using linear graphs. Learn about proportionality, constant rate of change, and how to identify proportional relationships in linear functions. Suitable for Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation