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To find the rate of change of the turkey's temperature after an hour, we need to determine the slope of the tangent line to the curve at \( t = 60 \) minutes. Here's how we can approximate this:

1. **Locate the Point:** At \( t = 60 \) minutes (point \( P \) on the graph), we identify the temperature value on the curve. From the graph, it looks like the temperature is approximately 120°F.

2. **Draw the Tangent:** The problem suggests using the slope of the tangent line at this point. On the graph, we look for a line that just touches the curve at \( t = 60 \) minutes and estimate the change in temperature over a small time interval.

3. **Estimate the Slope:** The slope of the tangent can be found using the formula:
   \[
   \text{slope} = \frac{\Delta T}{\Delta t},
   \]
   where \( \Delta T \) is the change in temperature and \( \Delta t \) is the change in time.

4. **Approximation:** By looking at the tangent line, we estimate the rate of temperature change. Suppose we approximate a fall of 20°F over a 50-minute interval around \( t = 60 \) (from approximately 140°F at \( t = 50 \) minutes to 120°F at \( t = 100 \) minutes).

   Then,
   \[
   \text{slope} \approx \frac{-20 \, \text{°F}}{50 \, \text{min}} = -0.4 \, \text{°F/min}.
   \]

So, the rate of change of the temperature after an hour is approximately **-0.4°F/min**.

Would you like a more detailed explanation or have any questions?

Here are some related questions for further exploration:
1. How does the rate of change vary at different times (e.g., after 30 minutes, 90 minutes)?
2. What is the long-term temperature of the turkey as time approaches infinity?
3. Can we model this cooling process using Newton's Law of Cooling?
4. How would a different room temperature affect the rate of cooling?
5. How can you use calculus to find the exact rate of change if the equation of the curve is known?

**Tip:** When estimating slopes on a graph, try to pick two points as far apart as possible on the tangent line to minimize error in the slope calculation.

By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour (in °F/min).

Learn how to estimate the rate of change of the temperature for a cooling turkey using the slope of the tangent line at t = 60 minutes. In this example, key calculus concepts like rate of change and the slope formula are explored. Suitable for students in Grades 10-12.

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