Chapter 3.4, Problem 30AYU

Which Model? A cricket makes a chirping noise by sliding its wings together rapidly. Perhaps you have noticed that the rapidity of chirps seems to increase with the temperature. The following data list the temperature (in degrees Fahrenheit) and the number of chirps per second for the striped ground cricket.

(a) Using a graphing utility, draw a scatter diagram of the data, treating temperature as the independent variable. What type of relation appears to exist between temperature and chirps per second?

(b) Based on your response to part (a), find either a linear or a quadratic model that best describes the relation between temperature and chirps per second.

Use your model to predict the chirps per second if the temperature is $80°\text{F}$.

To determine

To calculate:

Graph a scattered diagram and determine the type of relation that exists between the 2 variables.

Find the equation of best fit.

Predict the chirps per second if the temperature is ${80}^{\circ }F$.

Answer to Problem 30AYU

The graph is given below.

The equation of best fit is $C=0.2331x-2.0372$.

There’re around 16 chirps per second if the temperature is ${80}^{\circ }F$.

Explanation of Solution

Given:

The given data represents the relation between the temperature and the number of chirps per second for the striped ground cricket..

Formula used:

For a quadratic equation $y=a{x}^2+bx+c,$ if $a$ is positive, then the vertex $\left(\frac{-b}{2a},<mtext> </mtext>y\left(\frac{-b}{2a}\right)\right)$ is the minimum point and if $a$ is negative, the vertex $\left(\frac{-b}{2a},<mtext> </mtext>y\left(\frac{-b}{2a}\right)\right)$ is the maximum point.

Calculation:

We can draw the scatter diagram using Microsoft Excel.

Thus, on entering the $x$ and the $y$ values on excel, we have to choose the Scatter diagram form the insert option.

Then for getting the equation of best fit, we have to choose the Layout option and then Trendline and then more Trendline option. Then choose the option polynomial and then display the equation.

Thus, we get the scatter diagram with the equation of best fit as

a. The scatter diagram is drawn above and we can see that the given variables exhibit a linear relationship with positive slope.

b. The equation of best fit is $C=0.2331x-2.0372$.

c. When $x=80,$ we get

$C(80)=0.2331(80)-2.0372=16.3408$

There’re around 16 chirps per second if the temperature is ${80}^{\circ }F$