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Chapter 0: Problem 10

Entomologists have discovered that a linear relationship exists between the number of chirps of crickets of a certain species and the air temperature. When the temperature is \(70^{\circ} \mathrm{F}\), the crickets chirp at the rate of 120 times \(/ \mathrm{min}\); when the temperature is \(80^{\circ} \mathrm{F}\), they chirp at the rate of 160 times/min. a. Find an equation giving the relationship between the air temperature \(t\) and the number of chirps per minute, \(N\), of the crickets. b. Find \(N\) as a function of \(t\), and use this formula to determine the rate at which the crickets chirp when the temperature is \(102^{\circ} \mathrm{F}\).

Short Answer

Expert verified
The linear relationship between the air temperature, \(t\), and the number of chirps per minute, \(N\), is given by the equation \(N = 4t - 160\). When the temperature is \(102^{\circ} \mathrm{F}\), the crickets chirp at a rate of 248 times per minute.

Step by step solution

01

Determine the Slope and the y-intercept

We are given two points on the line: \((70, 120)\) and \((80, 160)\). First, find the slope (m) of the line using the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Then, use point-slope form to find the equation of the line, and finally, find the y-intercept (b).
02

Calculate the Slope

Using the slope formula and the given points \((70, 120)\) and \((80, 160)\): \[m = \frac{160 - 120}{80 - 70} = \frac{40}{10} = 4\]
03

Use the Point-Slope Form to Get the Equation

Now, use the point-slope form of a linear equation with one of the given points (let's use \((70, 120)\)) and the slope we found in Step 2: \[N - 120 = 4(t - 70)\]
04

Solve for the y-intercept (b)

Now, we'll solve the equation from Step 3 for \(N\) in terms of \(t\) and find the y-intercept (b) by putting it in y-intercept form, \(N = mt + b\): \[N = 4(t - 70) + 120\] \[N = 4t - 280 + 120\] \[N = 4t - 160\] So, the equation giving the relationship between the air temperature, \(t\), and the number of chirps per minute, \(N\), is: \[N = 4t - 160\]
05

Determine Chirp Rate at a Given Temperature

Now, we'll find the chirp rate when the temperature is \(102^{\circ} \mathrm{F}\) using the equation we found in Step 4. Plug in \(t = 102\) and find \(N\): \[N = 4(102) - 160\)\ \[N = 408 - 160\] \[N = 248\] The crickets chirp at the rate of 248 times per minute when the temperature is \(102^{\circ} \mathrm{F}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope of a Line
In statistics and mathematics, the slope of a line is a number that describes both the direction and the steepness of the line. The slope is often denoted by the letter 'm' and is calculated as the rise over run between two points on the line.

To find the slope using two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), we apply the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). In the context of the provided exercise, we saw that the slope represented the rate at which cricket chirps changed with respect to temperature changes. The two points given, \( (70, 120) \) and \( (80, 160) \), allowed us to calculate the slope as \( 4 \). This indicates that for every one degree rise in temperature, the number of chirps increases by four.

Understanding the concept of slope is crucial when studying linear relationships because it provides insight into how one variable responds to changes in another. It's also the foundation for determining the equation of a line, which is invaluable for predicting values.
Point-Slope Form
The point-slope form is a way of writing the equation of a line when you know the slope and one point on the line (\( (x_1, y_1) \)). This form is expressed as \( y - y_1 = m(x - x_1) \), where 'm' is the slope.

In our exercise, once the slope was determined to be 4, we used the point-slope form with the point \( (70, 120) \) to create the equation \( N - 120 = 4(t - 70) \). Through the point-slope equation, we can create a linear model that predicts the number of chirps (\( N \)) for any given temperature (\( t \)).

The simplicity of the point-slope form makes it especially useful when you have a specific point and slope and want to quickly form the equation of a line. It's a direct way to write the equation which can then be manipulated into other forms, such as slope-intercept form, depending on the application.
Y-intercept
The y-intercept is the point where the line crosses the y-axis of a graph. This point can be identified by the x-value being 0, leaving us with the y-value which is denoted by the letter 'b' in linear equations of the form \( y = mx + b \). The y-intercept is crucial because it provides a starting point for the line and, along with the slope, completely determines the position of the line on a graph.

In the solution to the exercise, the y-intercept was found by rearranging the point-slope equation into the slope-intercept form, giving us \( N = 4t - 160 \). Here, the number \( -160 \) represents the y-intercept, indicating that if the temperature were 0, theoretically, the number of chirps per minute would be at -160, which in this context highlights that the model only works within a certain range of temperatures. It also reflects that the y-intercept in this scenario may not have a practical biological interpretation, but it's essential for the mathematical model of the linear relationship.

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Most popular questions from this chapter

The number of driver fatalities due to car crashes, based on the number of miles driven, begins to climb after the driver is past age 65 years. Aside from declining ability as one ages, the older driver is more fragile. The number of driver fatalities per 100 million vehicle miles driven is approximately \(N(x)=0.0336 x^{3}-0.118 x^{2}+0.215 x+0.7 \quad 0 \leq x \leq 7\) where \(x\) denotes the age group of drivers, with \(x=0\) corresponding to those aged \(50-54\) years, \(x=1\) corresponding to those aged \(55-59, x=2\) corresponding to those aged \(60-64, \ldots\), and \(x=7\) corresponding to those aged \(85-89 .\) What is the driver fatality rate per 100 million vehicle miles driven for an average driver in the \(50-54\) age group? In the \(85-89\) age group?

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