Question

thought - provoking the table shows the temperatures, y (in degrees fahrenheit) of a cup of tea after x minutes. write a function that models the data and can be used to predict the temperature of the tea after 20 minutes. time, x: 0, 2, 4, 6, 8, 10; temperature, y: 190, 164, 146, 131, 120, 111. prediction: °f. explain your reasoning.

Explanation:

Step1: Assume the function is linear $y = mx + b$.

We use the first - two data points $(x_1,y_1)=(0,190)$ and $(x_2,y_2)=(2,164)$. The slope $m$ is calculated as $m=\frac{y_2 - y_1}{x_2 - x_1}$.
$m=\frac{164 - 190}{2-0}=\frac{- 26}{2}=-13$.

Step2: Find the y - intercept $b$.

When $x = 0$, $y=b$. From the point $(0,190)$, we know $b = 190$. So the linear function is $y=-13x + 190$.

Step3: Predict the temperature at $x = 20$.

Substitute $x = 20$ into the function $y=-13\times20 + 190$.
$y=-260+190=-70$. But a negative temperature doesn't make physical sense. Let's try an exponential decay model $y = a\cdot b^x$.
When $x = 0$, $y=a\cdot b^0=a$. From the point $(0,190)$, we have $a = 190$. So $y = 190\cdot b^x$.
Using the point $(2,164)$, we substitute $x = 2$ and $y = 164$ into $y = 190\cdot b^x$: $164=190\cdot b^2$. Then $b^2=\frac{164}{190}\approx0.8632$, and $b=\sqrt{\frac{164}{190}}\approx0.929$.
The function is $y = 190\cdot(0.929)^x$.

Step4: Predict the temperature at $x = 20$.

Substitute $x = 20$ into $y = 190\cdot(0.929)^{20}$.
$y = 190\times0.247\approx47$.

Answer:

$y = 190\cdot(0.929)^x$; Prediction: $47^{\circ}F$