Question

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Think of two real-life quantities that are related to each other and construct a function that models the relationship between the two quantities. The function should be a piecewise function where not all components are linear. Describe the scenario you are modeling, the definitions of the variables involved in the function, and a graph or equation of the function.

Answer 1

Let's consider a scenario involving the relationship between the age of a pet dog and its weight. Pet owners often notice that dogs grow rapidly during their early years and then their growth slows as they reach maturity. After maturity, weight might stabilize or even decrease slightly with age due to changes in diet or health conditions.

### Scenario: Dog Age and Weight

**Variables:**
- Let \( x \) represent the age of the dog in years.
- Let \( w \) represent the weight of the dog in kilograms.

### Piecewise Function

The piecewise function modeling this relationship could be defined as follows:

\[
w(x) =
\begin{cases}
1.5x + 5 & \text{if } 0 \leq x < 2 \\
15 & \text{if } 2 \leq x < 7 \\
-1.5x + 25 & \text{if } 7 \leq x \leq 15
\end{cases}
\]

### Explanation of Each Piece:
1. **First Component \( 1.5x + 5 \) (for \( 0 \leq x < 2 \))**:
- In the first two years, a puppy grows rapidly. This linear function represents the increasing weight as the puppy ages. The constant term (5 kg) represents the average weight of a small puppy when it is born.

2. **Second Component \( 15 \) (for \( 2 \leq x < 7 \))**:
- After about 2 years, many dogs reach their adult weight and maintain it. Thus, for ages 2 to 7 years, the weight stabilizes around 15 kg. This constant function indicates that the dog has reached its maturity and weight does not change significantly during this period.

3. **Third Component \( -1.5x + 25 \) (for \( 7 \leq x \leq 15 \))**:
- After about 7 years, many dogs begin to gradually lose weight, especially larger breeds. This linear function represents the weight decline as the dog ages into its senior years, indicating the effects of aging and possibly health issues. The weight starts at about 25 kg when the dog turns 7 years old.

### Graph of the Function
To visualize this function, we can plot the relationship—note that the transitions between the segments will be at the specified ages.

1. For ages 0-2, the graph is a line with a positive slope.
2. At age 2, the graph jumps to a constant line at 15 kg.
3. At age 7, the graph begins to slope downward, indicating weight loss.

This piecewise relationship highlights how a dog’s weight changes with age, incorporating different growth phases and effects of aging.

### Conclusion
This model gives pet owners insights into expected changes in their dog's weight as they age, helping them monitor their pet's health appropriately.