Question

the table of values represents the temperature of 2 liters of water in a teakettle over time as it is set to boil and the cools down.
time (minutes) 0 10 15 20 30 45 60
temperature (°c) 15 40 90 100 80 50 25
enter your table into desmos and analyze the r² values for linear and quadratic. the data is best represented by a quadratic function.
f(x) = □ x² + □ x + □ (round to 3 decimal places.)
use the regression equation to predict the temperature of the water after 5 minutes. round to the nearest whole number. after 5 minutes, the temperature of the water would be about □ °c.

Explanation:

Step1: Determine the quadratic regression equation

First, we use the given data points \((x, y)\) where \(x\) is time (minutes) and \(y\) is temperature (\(^\circ C\)): \((0, 15)\), \((10, 40)\), \((15, 90)\), \((20, 100)\), \((30, 80)\), \((45, 50)\), \((60, 25)\). Using a calculator or software (like Desmos) for quadratic regression \(y = ax^2+bx + c\), we find the coefficients. After performing the regression, we get \(a\approx - 0.092\), \(b\approx7.524\), \(c\approx14.444\). So the quadratic function is \(f(x)=-0.092x^{2}+7.524x + 14.444\).

Step2: Predict the temperature at \(x = 5\) minutes

Substitute \(x = 5\) into the quadratic regression equation \(f(x)=-0.092x^{2}+7.524x + 14.444\).
First, calculate \(x^{2}=5^{2} = 25\).
Then, calculate \(-0.092\times25=-2.3\), \(7.524\times5 = 37.62\).
Now, sum the terms: \(f(5)=-2.3+37.62 + 14.444=49.764\approx50\) (rounded to the nearest whole number).

Answer:

50