Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Cricket Chirping At \(68^{\circ} \mathrm{F}\), a certain species of cricket chirps 112 times per minute. At \(46^{\circ} \mathrm{F}\), the same cricket chirps 24 times per minute. (a) Express the number of chirps, \(y,\) as a linear function of the Fahrenheit temperature. (b) If the temperature is \(60^{\circ} \mathrm{F}\), how many times will the cricket chirp per minute? (c) If you count the number of cricket chirps in one-half minute and hear 40 chirps, what is the temperature?

Step-by-Step Solution

Answer

(a) \(y = 4x - 160\); (b) 80 chirps; (c) \(60^{\circ} \mathrm{F}\).

Step 1: Understand the Problem

We are given that at two different temperatures, the number of cricket chirps per minute is different. We need to express the number of chirps, \(y\), as a linear function of temperature \(x\), solve for the number of chirps at \(60^{\circ} \mathrm{F}\), and find the temperature when we hear 40 chirps in half a minute.

Step 2: Find the Slope (m)

The formula for the slope \(m\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((68, 112)\) and \((46, 24)\), we find:\[m = \frac{112 - 24}{68 - 46} = \frac{88}{22} = 4.\]

Step 3: Write the Linear Equation

The linear equation is of the form \(y = mx + b\). Using the slope \(m = 4\) and the point-slope form with \((68, 112)\), we can determine \(b\):\[112 = 4(68) + b \112 = 272 + b \b = -160.\]So, the linear equation is \(y = 4x - 160\).

Step 4: Solve for the Chirping at 60°F

Substitute \(x = 60\) into the linear equation \(y = 4x - 160\):\[y = 4(60) - 160 = 240 - 160 = 80.\]Thus, the cricket chirps 80 times per minute at \(60^{\circ} \mathrm{F}\).

Step 5: Calculate Temperature from Chirping Rate

If the number of chirps in half a minute is 40, then the rate per minute is \(40 \times 2 = 80\) chirps per minute. Substitute \(y = 80\) into the equation \(y = 4x - 160\):\[80 = 4x - 160 \240 = 4x \x = 60.\]The temperature is \(60^{\circ} \mathrm{F}\).

Key Concepts

Slope

The slope of a linear function is a measure of how steep the line is and how quickly it rises or falls. Mathematically, it is represented as the change in the y-values (such as the number of chirps) divided by the change in the x-values (such as the temperature).

• In the formula for the slope, denoted as \( m \), we use \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• This calculation gives us the rate of change between the two variables.

To find the slope in our cricket chirping problem, we used two given points:

• The number of chirps at \(68^{\circ} \mathrm{F}\) is 112, and at \(46^{\circ} \mathrm{F}\) it is 24.
• Using these points \( (68, 112) \) and \( (46, 24) \), we calculate the slope as \( \frac{88}{22} = 4 \).

This tells us that for every increase of \(1^{\circ} \mathrm{F}\) in temperature, there are 4 more chirps per minute.

Temperature Calculation

Temperature calculation is a crucial part of understanding how variables like chirping rates change with temperature. In our problem, we express the number of chirps as a function of temperature. The relationship is linear, meaning it can be represented as a straight line on a graph.

• Using the linear equation \( y = 4x - 160 \), you can calculate the chirping frequency for any given temperature.
• For example, at \(60^{\circ} \mathrm{F}\), substituting \(x\) with 60 provides us \(y = 4(60) - 160 = 80\). The cricket chirps 80 times per minute at this temperature.

This type of calculation helps us predict cricket chirps for any temperature, offering a deeper understanding of the correlation between these two variables. Maintaining accuracy in such predictions demonstrates the practical application of linear equations in everyday scenarios.

Linear Equation

A linear equation is an algebraic expression that represents a straight line when graphed. For our cricket chirping problem, the linear equation is

• \( y = 4x - 160 \).

In this equation:

• \(y\) represents the number of chirps per minute,
• \(x\) is the temperature in Fahrenheit,
• \(4\) is the slope, showing that chirping rate increases by 4 for each degree of temperature rise.

To find the y-intercept \(b\), which is -160 in this case, we used the slope-intercept form \(y = mx + b\) with known data points. This intercept indicates the hypothetical chirping rate if the temperature could be \(0^{\circ} \mathrm{F}\), allowing us to prepare predictions and understand the function's behavior across a potential range.By solving the linear equation, we can also determine unknown temperatures from a given chirping rate, showcasing how versatile and informative linear functions can be.