Experimental Setup:
- A hot cup of tea is placed on a table.
- The tea’s temperature is recorded every 5 minutes for 2 hours.
- The room temperature is measured to be 24 °C.
Instructions:
1. Using Desmos. Copy the data provided and paste it into the box below. (A table with the pasted data and headings 𝑥1 and 𝑦1 should appear.)
3. Perform a regression for each of the models below. Note that (i) the regression formulas given below assume your data table has column headings 𝒙𝟏 and 𝒚𝟏, and (ii) the subscripts 1, 2, and 3 on the constants 𝒂, 𝒃, 𝒄, 𝒅 are necessary so that Desmos can distinguish between them.
| Type of model | Regression formula to enter in Desmos |
|---|---|
| Quadratic | y1~a1x1^2+b1x1+c1 |
| Rational | y1~a2x1+b2/c2x1+d2 |
| Exponential | y1~a3b3^k(x1-d3)+c3 |
4. Use the “Parameters” given for each regression to state the equation of each model in the table below. Round each parameter to two decimal places. (If doing so causes a parameter to round to zero, round to keep one nonzero digit, instead. E.g., 0.0004621 would round to 0.00, so round to 0.0005 instead.)
| Type of Model | Equation |
|---|---|
| Quadratic | |
| Rational | |
| Exponential |
5. What is the domain of all three of these models?
6. Compare and contrast the three models. Which model fits the data the best? (I.e., Which is the best model
for 0 ≤ 𝑡 ≤ 120 ?) Explain how you know your chosen model is more accurate than the other two.
7. Which model will give the best predictions of the temperature after 120 min? (I.e., Which is the best model for 𝑡 > 120 ?) Explain how you know your chosen model is more accurate than the other two.
8. Determine the room temperature according to your rational model, rounded to one decimal place.
9. Determine the room temperature according to your exponential model, rounded to one decimal place.
10. Based on all the above considerations, which model do you think is the most appropriate fit for the data? Explain.
