Question

At 108°F, a certain insect chirps at a rate of 60 times per minute, and at 113°F, they chirp 105 times per minute. Write an equation in slope-intercept form that represents the situation.

Answer 1

the equation in slope-intercept form representing the relationship between the temperature (x) and the chirping rate (y) of the insect is:

`\[y = 9x - 912\]`

Step-by-step explanation:

To create an equation in slope-intercept form (y = mx + b) that represents the relationship between the temperature (x) and the chirping rate (y) of the insect, we can use the two data points provided: (108°F, 60 chirps per minute) and (113°F, 105 chirps per minute).

First, we can determine the slope (m) using the formula for slope:

`\[m = \frac{{\text{change in } y}}{{\text{change in } x}}\]`

`\[m = \frac{{105 - 60}}{{113 - 108}} = \frac{{45}}{{5}} = 9\]`

Now that we have the slope (m), we can use the point-slope form of a linear equation to find the equation of the line:

`\[y - y_1 = m(x - x_1)\]`

Let's choose one of the points, say (108°F, 60 chirps per minute), and substitute it into the equation along with the slope.

Using the point (108, 60):

`\[y - 60 = 9(x - 108)\]`

Now, let's simplify and solve for y to get the equation in slope-intercept form (y = mx + b):

`\[y - 60 = 9x - 972\]`

`\[y = 9x - 972 + 60\]`

`\[y = 9x - 912\]`

Therefore, the equation in slope-intercept form representing the relationship between the temperature (x) and the chirping rate (y) of the insect is:

`\[y = 9x - 912\]`