COOKING The amount of time it takes to cook a turkey increases with the weight of the turkey. It is recommended that you cook a \( 10-1 b \) turkey for 3 hours. An extra 12 minutes of cooking time is necessary for each additional pound of turkey. Is the cook time proportional to the weight of the turkey? Explain your reasoning.

No, the cook time is **not** proportional to the weight of the turkey. Here's why: ### Understanding Proportionality In a **proportional relationship**, one quantity is a constant multiple of another. Mathematically, this means: \[ \text{Cook Time} = k \times \text{Weight} \] where \( k \) is a constant. In such a relationship, if you double the weight, the cook time also doubles. ### Given Situation - **Base Cook Time:** A turkey of a certain base weight (let's assume \( W_0 \)) takes **3 hours** to cook. - **Additional Time:** For each extra pound beyond \( W_0 \), you add **12 minutes**. This can be expressed as: \[ \text{Cook Time} = 3 \text{ hours} + 12 \text{ minutes} \times (\text{Weight} - W_0) \] ### Why It's Not Proportional - **Fixed Base Time:** The initial 3-hour cook time does not scale with the weight of the turkey. It's a fixed component regardless of the turkey's size. - **Additional Time is Proportional:** Only the extra time beyond the base weight increases proportionally with each additional pound. Because of the fixed base time, the overall relationship between cook time and weight includes both a constant and a variable component. This means the cook time increases linearly with weight but **not proportionally**. In other words, the ratio of cook time to weight isn't constant across different weights. ### Visual Representation Imagine plotting cook time against weight: - **Proportional Relationship:** A straight line through the origin (0,0). - **Given Scenario:** A straight line that starts at ( \( W_0 \), 3 hours ) and then increases linearly, which **does not** pass through the origin. ### Conclusion The presence of a fixed base cook time means that while cook time does increase with weight, it doesn't do so in a directly proportional manner. **Answer:** No. Because a fixed base time is required and additional time depends on weight, the cook time increases linearly but does not scale proportionally with the turkey’s weight.