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Let's analyze the problem step by step.

### 1. **Identifying Variables**
   - \( x \): The number of chirps per minute.
   - \( y \): The temperature in degrees Fahrenheit.

### 2. **Given Information**
   The problem gives two points:
   - When the crickets chirp 124 times per minute, the temperature is 68°F. This gives the point \((124, 68)\).
   - When the crickets chirp 172 times per minute, the temperature is 80°F. This gives the point \((172, 80)\).

### 3. **Finding the Slope**
   The slope \( m \) of the line can be found using the slope formula:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Substituting the values of the two points \((124, 68)\) and \((172, 80)\):
   \[
   m = \frac{80 - 68}{172 - 124} = \frac{12}{48} = 0.25
   \]

   So, the slope \( m = 0.25 \).

### 4. **Finding the y-Intercept**
   Now, we can use the point-slope form of the equation of a line:
   \[
   y - y_1 = m(x - x_1)
   \]
   Using the point \((124, 68)\) and the slope \( m = 0.25 \):
   \[
   y - 68 = 0.25(x - 124)
   \]
   Simplifying this:
   \[
   y - 68 = 0.25x - 31
   \]
   \[
   y = 0.25x - 31 + 68
   \]
   \[
   y = 0.25x + 37
   \]
   So, the equation of the line is:
   \[
   y = 0.25x + 37
   \]
   This gives us the y-intercept of \( 37 \).

### 5. **Answering the Questions**
   - The **y-intercept** is 37.
   - The **slope** is 0.25.
   
### Part a: Equation
The equation for the line modeling the situation is:
\[
y = 0.25x + 37
\]

### Part b: Finding the Temperature When Chirping 150 Times per Minute
We can substitute \( x = 150 \) into the equation:
\[
y = 0.25(150) + 37 = 37.5 + 37 = 74.5
\]
So, when the crickets chirp 150 times per minute, the temperature is approximately **74.5°F**.

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### Related Questions:
1. What does the slope of the line represent in the context of this problem?
2. How would the equation change if the crickets chirped 200 times per minute at 90°F?
3. How would you graph this equation and what would the graph look like?
4. If the temperature was 85°F, how many chirps per minute would the crickets make?
5. What is the significance of the y-intercept in this situation?

### Tip:
The slope in linear relationships can often be interpreted as the rate of change between two variables—in this case, the rate at which the temperature changes per additional cricket chirp.

Biologists have found that the number of chirps some crickets make per minute is related to temperature. When crickets chirp 124 times per minute, it is about 68°F. When they chirp 172 times per minute, it is about 80°F. a) Find an equation for the line that models this situation. b) How warm is it when the crickets are chirping 150 times a minute?

Learn how to model the relationship between cricket chirps per minute and temperature using a linear equation. This problem involves finding the slope and y-intercept of a line, then using it to predict the temperature when crickets chirp 150 times a minute.

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