https://file.aiquickdraw.com/mathbot/file/1729616713272e05gp4k5.jpg

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!

Find the linear or exponential equation of best fit for the given time-temperature data. Then, estimate the temperature of the liquid after 1 minute (60 seconds).

Discover how to find the linear equation of best fit for a dataset showing time in seconds and temperature in degrees. This step-by-step solution covers slope calculation and the y-intercept, with an estimate for temperature after 60 seconds. Ideal for Grades 9-11 students learning algebra and linear modeling.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation