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68 A container of food was moved from a standard freezer to a deep freezer to preserve the food for a longer period of time. The temperature of the food was modeled by an exponential function. Can we use this model to predict the food's temperature after 20 minutes? f (x) Temperature (in Fahrenheit) = 15 -14 13 12 11 10 286 9 6 10 5 4 2 1 - 0.22e0.23x + 14 with r² = 0.90 3²0.90 0 f(x)=-0.22e0.23x +14 2 4 6 8 ● 10 Minutes ● 12 14 Yes, we can use this model because r² is close to 1, and x=20 is a safe extrapolation. No, we cannot use this model because the trend does not continue past x=18. No, we cannot use this model because x=20 is an extreme extrapolation. 16 Yes, we can use this model because the data follows an exponential trend. 20

68 A container of food was moved from a standard freezer to a deep freezer to preserve the food for a longer period of time. The temperature of the food was modeled by an exponential function. Can we use this model to predict the food's temperature after 20 minutes? f (x) Temperature (in Fahrenheit) = 15 -14 13 12 11 10 286 9 6 10 5 4 2 1 - 0.22e0.23x + 14 with r² = 0.90 3²0.90 0 f(x)=-0.22e0.23x +14 2 4 6 8 ● 10 Minutes ● 12 14 Yes, we can use this model because r² is close to 1, and x=20 is a safe extrapolation. No, we cannot use this model because the trend does not continue past x=18. No, we cannot use this model because x=20 is an extreme extrapolation. 16 Yes, we can use this model because the data follows an exponential trend. 20

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**Transcription and Explanation:** **Title:** Predicting Food Temperature in a Deep Freezer Using an Exponential Model **Introduction:** A container of food was moved from a standard freezer to a deep freezer to preserve the food for a longer period. The temperature of the food was modeled by an exponential function. The question raised is whether we can use this model to predict the food's temperature after 20 minutes. **Exponential Function Model:** - The function used is: \( f(x) = -0.22e^{0.23x} + 14 \) - The coefficient of determination is \( r^2 = 0.90 \). **Graph Explanation:** - **Axes:** The x-axis represents time in minutes, ranging from 0 to 20. The y-axis represents temperature in Fahrenheit, ranging from 0 to 15. - **Data Points:** Blue dots plotted to indicate recorded temperatures at specific time intervals. - **Exponential Curve:** A red curve is fitted to the data points, demonstrating the exponential decline in temperature over time. **Options for Model Usage:** 1. Yes, we can use this model because \( r^2 \) is close to 1, and \( x = 20 \) is a safe extrapolation. 2. No, we cannot use this model because the trend does not continue past \( x = 18 \). 3. No, we cannot use this model because \( x = 20 \) is an extreme extrapolation. 4. Yes, we can use this model because the data follows an exponential trend. **Conclusion:** This model provides a mathematical representation of how the temperature decreases when the food is transferred to a deep freezer. The decision to use this model for extrapolation depends on the trust in the model's fit and the appropriateness of predicting temperatures beyond the observed data range.
Transcribed Image Text:**Transcription and Explanation:** **Title:** Predicting Food Temperature in a Deep Freezer Using an Exponential Model **Introduction:** A container of food was moved from a standard freezer to a deep freezer to preserve the food for a longer period. The temperature of the food was modeled by an exponential function. The question raised is whether we can use this model to predict the food's temperature after 20 minutes. **Exponential Function Model:** - The function used is: \( f(x) = -0.22e^{0.23x} + 14 \) - The coefficient of determination is \( r^2 = 0.90 \). **Graph Explanation:** - **Axes:** The x-axis represents time in minutes, ranging from 0 to 20. The y-axis represents temperature in Fahrenheit, ranging from 0 to 15. - **Data Points:** Blue dots plotted to indicate recorded temperatures at specific time intervals. - **Exponential Curve:** A red curve is fitted to the data points, demonstrating the exponential decline in temperature over time. **Options for Model Usage:** 1. Yes, we can use this model because \( r^2 \) is close to 1, and \( x = 20 \) is a safe extrapolation. 2. No, we cannot use this model because the trend does not continue past \( x = 18 \). 3. No, we cannot use this model because \( x = 20 \) is an extreme extrapolation. 4. Yes, we can use this model because the data follows an exponential trend. **Conclusion:** This model provides a mathematical representation of how the temperature decreases when the food is transferred to a deep freezer. The decision to use this model for extrapolation depends on the trust in the model's fit and the appropriateness of predicting temperatures beyond the observed data range.
A company has limited releases of RPG (Role-Playing Game) sourcebooks each month. They track the total number of books sold each day after the release date each month. They have a total supply of 1000 books each month, so they cannot sell more than that. They wish to determine how many books will have sold by 13 days since release. Which of the following statements is true? Graph Explanation: - The graph displays the relationship between the "Number of Books Sold" (y-axis) and "Days since Release" (x-axis). - The curve shown on the graph suggests an exponential growth pattern in the number of books sold over time. - The coefficient of determination (R²) is 0.98, indicating a very strong fit of the model to the observed data points. - The data points on the graph are plotted as blue dots and depict the actual sales data for several days. Statements: - ⃝ This is a strong model, so the number of books sold 13 days since release is within the safe extrapolation range. - ⃝ Even though this is a strong model, it is more appropriate to use an exponential model, so a prediction should not be made using this model. - ⃝ This is a moderate model, so the number of books sold 13 days since release is not within the safe extrapolation range. - ⃝ The model is not appropriate to use beyond 12 days since release.
Transcribed Image Text:A company has limited releases of RPG (Role-Playing Game) sourcebooks each month. They track the total number of books sold each day after the release date each month. They have a total supply of 1000 books each month, so they cannot sell more than that. They wish to determine how many books will have sold by 13 days since release. Which of the following statements is true? Graph Explanation: - The graph displays the relationship between the "Number of Books Sold" (y-axis) and "Days since Release" (x-axis). - The curve shown on the graph suggests an exponential growth pattern in the number of books sold over time. - The coefficient of determination (R²) is 0.98, indicating a very strong fit of the model to the observed data points. - The data points on the graph are plotted as blue dots and depict the actual sales data for several days. Statements: - ⃝ This is a strong model, so the number of books sold 13 days since release is within the safe extrapolation range. - ⃝ Even though this is a strong model, it is more appropriate to use an exponential model, so a prediction should not be made using this model. - ⃝ This is a moderate model, so the number of books sold 13 days since release is not within the safe extrapolation range. - ⃝ The model is not appropriate to use beyond 12 days since release.
Expert Solution
Step 1: Question 68

The coefficient of determination (r2) for the model is 0.90, which means 90% of the variation in temperature of food can be explained by the fitted exponential model. The model is strong. But x=20 is very extreme, as it lies way-beyond the range of x and will given a very extreme prediction which is highly likely to be not true.


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