Let's start by finding the linear equation for the model based on the given data.

The data points for time (in seconds) and temperature (in degrees) are:

• $(15, 80)$
• $(20, 75)$
• $(30, 68)$
• $(40, 63)$
• $(45, 56)$

We will first find the linear equation of best fit, which has the form $y = mx + b$, where:

• $y$ is the temperature,
• $x$ is the time in seconds,
• $m$ is the slope, and
• $b$ is the y-intercept.

Step 1: Find the slope $m$

The slope $m$ is calculated by using the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's calculate the slope between two points, say $(15, 80)$ and $(45, 56)$:

$m = \frac{56 - 80}{45 - 15} = \frac{-24}{30} = -0.8$

Step 2: Find the y-intercept $b$

Now that we have the slope $m = -0.8$, we can use one of the points, say $(15, 80)$, to find the y-intercept $b$:

$80 = -0.8(15) + b$ $80 = -12 + b$ $b = 92$

Step 3: Write the equation

Thus, the linear equation of best fit is:

$y = -0.8x + 92$

Step 4: Estimate the temperature after 1 minute

1 minute = 60 seconds. Plugging $x = 60$ into the equation:

$y = -0.8(60) + 92 = -48 + 92 = 44$

Thus, the estimated temperature after 1 minute is $44^\circ$.

Would you like more details on any of the steps?

Here are some related questions for further exploration:

1. How do you calculate the slope between any two points?
2. What is the significance of the y-intercept in this context?
3. How would you check if an exponential model is a better fit for the data?
4. How can you verify the accuracy of a linear regression model?
5. Can a quadratic or polynomial model improve the fit for this data set?

Tip: Always check if your data fits the chosen model well by plotting or calculating the residuals!