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.

Would you like any more details or explanations?

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.