Questions: The rate at which a cricket chirps depends on the temperature of the surrounding air. It is possible to estimate the air temperature by counting chirps. Examine the table. Use the table to answer parts (a) through (d). Temperature ( ° F ) Rate (chirps per minute) 50 43 60 86 70 129 80 172 90 215 a. Let g(F) be the number of chirps per minute a cricket makes when the temperature is F degrees Fahrenheit. Find an equation of g . The line should be found by interpolating between the first and last points. Verify that the graph of the equation comes close to the points in the scattergram of the data. g(F)=4.3 F-172 (Type your answer in slope-intercept form.) Does the graph of the equation come close to the scattergram of the data? No Yes b. Find g(51). What does it mean in this situation? g(51)= (Type an integer or a decimal.)

The rate at which a cricket chirps depends on the temperature of the surrounding air. It is possible to estimate the air temperature by counting chirps. Examine the table. Use the table to answer parts (a) through (d).

Temperature ( ° F ) Rate (chirps per minute)

50 43

60 86

70 129

80 172

90 215

a. Let g(F) be the number of chirps per minute a cricket makes when the temperature is F degrees Fahrenheit. Find an equation of g . The line should be found by interpolating between the first and last points. Verify that the graph of the equation comes close to the points in the scattergram of the data.

g(F)=4.3 F-172 (Type your answer in slope-intercept form.)

Does the graph of the equation come close to the scattergram of the data?

No

Yes

b. Find g(51). What does it mean in this situation?

g(51)= (Type an integer or a decimal.)

Transcript text: The rate at which a cricket chirps depends on the temperature of the surrounding air. It is possible to estimate the air temperature by counting chirps. Examine the table. Use the table to answer parts (a) through (d). Temperature ( ${ }^{\circ} \mathrm{F}$ ) & Rate (chirps per minute) 50 & 43 60 & 86 70 & 129 80 & 172 90 & 215 a. Let $g(F)$ be the number of chirps per minute a cricket makes when the temperature is $F$ degrees Fahrenheit. Find an equation of g . The line should be found by interpolating between the first and last points. Verify that the graph of the equation comes close to the points in the scattergram of the data. $g(F)=4.3 \mathrm{~F}-172$ (Type your answer in slope-intercept form.) Does the graph of the equation come close to the scattergram of the data? No Yes b. Find $g(51)$. What does it mean in this situation? $g(51)=$ $\square$ (Type an integer or a decimal.)

Solution

Solution Steps

Solution Approach

a. To find the equation of $ g(F) $, we need to determine the slope of the line that interpolates between the first and last points in the table. The slope $ m $ is calculated using the formula $ m = \frac{y_2 - y_1}{x_2 - x_1} $. Once we have the slope, we can use the point-slope form to find the equation of the line. Verify the equation by checking if it approximates the given data points.

b. To find $ g(51) $, substitute $ F = 51 $ into the equation of $ g(F) $ obtained in part (a). This will give the estimated number of chirps per minute at 51 degrees Fahrenheit.

Step 1: Determine the Equation of $ g(F) $

To find the equation of $ g(F) $, we first calculate the slope $ m $ using the points $ (50, 43) $ and $ (90, 215) $:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{215 - 43}{90 - 50} = \frac{172}{40} = 4.3 \]

Next, we find the y-intercept $ b $ using the point-slope form of the line:

\[ b = y_1 - m \cdot x_1 = 43 - 4.3 \cdot 50 = 43 - 215 = -172 \]

Thus, the equation of $ g(F) $ is:

\[ g(F) = 4.3F - 172 \]

Step 2: Verify the Equation

We can verify that the equation approximates the data points by substituting the values of $ F $ from the table into the equation and checking if the results are close to the corresponding chirp rates.

Step 3: Calculate $ g(51) $

Now, we substitute $ F = 51 $ into the equation to find $ g(51) $:

\[ g(51) = 4.3 \cdot 51 - 172 = 219.3 - 172 = 47.3 \]

Step 4: Interpret the Result

The value $ g(51) = 47.3 $ indicates that at a temperature of $ 51 \, ^\circ \mathrm{F} $, the cricket is estimated to chirp at approximately $ 47.3 $ chirps per minute.