To find the linear equation that models the temperature $T$ as a function of the number of chirps per minute $N$, we need to determine the slope-intercept form of the equation, which is generally given by:

$T(N) = mN + b$

Where:

• $m$ is the slope of the line
• $b$ is the y-intercept (the temperature when the number of chirps per minute $N$ is 0)

Step 1: Calculate the slope $m$

The slope $m$ can be calculated using the formula:

$m = \frac{\Delta T}{\Delta N} = \frac{T_2 - T_1}{N_2 - N_1}$

Given:

• $(N_1, T_1) = (120, 70)$
• $(N_2, T_2) = (171, 85)$

Substitute these values into the slope formula:

$m = \frac{85 - 70}{171 - 120} = \frac{15}{51} \approx 0.294$

Step 2: Use the slope to find the y-intercept $b$

Now, use the slope $m$ and one of the points to find the y-intercept $b$. We can use the point $(120, 70)$:

$70 = 0.294 \times 120 + b$

Solve for $b$:

$70 = 35.28 + b \implies b = 70 - 35.28 = 34.72$

Step 3: Write the final equation

Now that we have both $m$ and $b$, we can write the linear equation:

$T(N) = 0.294N + 34.72$

This is the linear equation that models the temperature $T$ as a function of the number of chirps per minute $N$.

Do you want any more details or have questions? Here are some related questions:

1. How would the equation change if the chirps per minute at 85°F were slightly different?
2. What does the slope of 0.294 represent in the context of this problem?
3. If a cricket chirps 150 times per minute, what would the temperature be according to this model?
4. How would you check the accuracy of this model with additional data points?
5. What is the significance of the y-intercept in this model?

Tip: When finding the slope of a line, remember it represents the rate of change between the two variables in your context—in this case, how much the temperature changes for each additional chirp per minute.