https://file.aiquickdraw.com/mathbot/file/1726795867687om3run47.jpg

I have reviewed the image and can help solve the problems based on the provided data and questions. Let's break down and address each part step by step.

### (a) Why might it make sense to fit the data using a quadratic function?

The data shows a pattern where the percentage of kernels popped increases as the temperature rises, then decreases at higher temperatures. This type of behavior — first increasing and then decreasing — suggests a parabolic relationship, which is typical for quadratic functions. A quadratic function can model such situations where there is an optimal point, followed by a decline.

### (b) Finding the quadratic regression equation

To solve this, I will perform a quadratic regression using the provided temperature and percentage data to find the equation \( y = ax^2 + bx + c \). I'll calculate the values for \( a \), \( b \), and \( c \).

Let me compute that for you.The quadratic regression equation that best fits the data is:

\[
y = -0.00398x^2 + 2.87676x - 427.96154
\]

Where:
- \( a = -0.00398 \)
- \( b = 2.87676 \)
- \( c = -427.96154 \)

### (c) Finding the temperature that produces the greatest percentage of kernels popped

To find the temperature that produces the greatest percentage of popped kernels, we need to find the vertex of the quadratic equation. The vertex occurs at:

\[
x = \frac{-b}{2a}
\]

Let me calculate that for you.The temperature that will produce the greatest percentage of kernels popped is approximately **362°F** (rounded to the nearest degree).

### (d) Expected number of popped kernels at the optimal temperature

If a typical batch consists of 800 kernels, we can estimate the number of popped kernels at the optimal temperature (362°F) by calculating the percentage popped at this temperature using the quadratic equation. Let me compute that next.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can compute the expected number of popped kernels by applying the quadratic equation we derived:

\[
y = -0.00398x^2 + 2.87676x - 427.96154
\]

Substitute \( x = 362 \) into the equation to find the percentage of popped kernels, then calculate:

\[
\text{popped kernels} = \frac{\text{percentage popped}}{100} \times 800
\]

Let me know if you would like more details on any of the steps or if you have additional questions.

---

Here are some questions to consider:
1. How would you interpret the results of the quadratic regression in a real-world context?
2. What other factors might influence the percentage of popped kernels?
3. Could a different type of regression (e.g., cubic) give a better fit?
4. What other data points would you collect if you were conducting this experiment?
5. How would temperature fluctuations affect the reliability of this data?

**Tip:** When performing regression, always ensure the chosen model fits the behavior of the data (e.g., increasing then decreasing for quadratic).

Based on temperature and percentage data, find the quadratic regression equation that best fits the data, calculate the temperature for the highest percentage of kernels popped, and estimate the number of popped kernels at that temperature.

Learn how to use quadratic regression to fit temperature and percentage data, calculate the optimal temperature for maximum popped kernels, and estimate the number of popped kernels at that temperature. Suitable for students in Grades 10-12.

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