Problem 81 - Introductory and Intermediate Algebra for College Students 4th

Question

Excited about the success of celebrity stamps, post office officials were rumored to have put forth a plan to institute two new types of thermometers. On these new scales, \(^{\circ} E\) represents degrees Elvis and \(^{\circ} M\) represents degrees Madonna. If it is known that \(40^{\circ} E=25^{\circ} M, 280^{\circ} E=125^{\circ} M\) and degrees Elvis is linearly related to degrees Madonna, write an equation expressing \(E\) in terms of \(M\).

Step-by-Step Solution

Answer

The equation expressing \(E\) in terms of \(M\) is \(E = ((125 - 25) / (280 - 40))*M + (40 - ((125 - 25) / (280 - 40))*25)\).

Step 1: Calculate the slope (m)

The slope of the linear relationship can be calculated by taking the difference between the \(M\) values and dividing it by the difference in \(E\) values. Hence, \(m = (125 - 25) / (280 - 40) \).

Step 2: Calculate the y-intercept (b)

After finding the slope, one can rewrite \(E = m*M + b\) as \(b = E - m*M\). Substituting one pair of \(E\) and \(M\) values into this equation, we find that \(b = 40 - m*25\). We recall from the previous step that \(m = (125 - 25) / (280 - 40)\). Hence, \(b = 40 - ((125 - 25) / (280 - 40))*25\).

Step 3: Write the equation

The linear equation describing the relationship between \(E\) and \(M\) is found in the form \(E = m*M + b\). Substituting the values for the slope and the y-intercept from the previous steps, the resulting equation is \(E = ((125 - 25) / (280 - 40))*M + (40 - ((125 - 25) / (280 - 40))*25)\).

Key Concepts

Slope Calculation

Calculating the slope is an essential part of understanding the relationship between two variables that are linearly related, like degrees Elvis and degrees Madonna in this exercise. The slope is a measure of how steeply one variable changes with respect to another. In this context, it helps us determine how changes in degrees Madonna (M) affect changes in degrees Elvis (E).To compute the slope, we use the formula for two given points: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, the points are represented as \((280, 125)\) and \((40, 25)\), where the first coordinate in each pair represents Elvis degrees and the second coordinate represents Madonna degrees. Pluging in these values:\( m = \frac{125 - 25}{280 - 40} \)Simplify the expression to find the slope:\( m = \frac{100}{240} = \frac{5}{12} \)This slope means that for every 12 degree increase in Madonna, there is a 5 degree increase in Elvis.

Equation Formulation

Once we have the slope, the next step in formulating the equation of a line is to identify the 'y-intercept'. In this linear equation, the y-intercept represents the value of Elvis degrees when Madonna degrees is zero. The equation of a line in slope-intercept form is:\[ E = m \cdot M + b \]where \(m\) is the slope and \(b\) is the y-intercept. To find \(b\), we can use one of the given points, such as \((40, 25)\). Substitute \( m = \frac{5}{12} \) and the point values into the equation:\[ 40 = \frac{5}{12} \cdot 25 + b \]Solving for \(b\), rearrange the equation:\[ b = 40 - \frac{5}{12} \cdot 25 \]Calculate the expression:\[ b = 40 - 10.4167 \approx 29.5833 \]Thus, the linear relationship or equation that defines the conversion between degrees Elvis and degrees Madonna is:\[ E = \frac{5}{12} M + 29.5833 \]

Temperature Conversion Scales

Understanding temperature conversion scales may seem daunting at first, but they're simply different systems for measuring temperature. In this exercise, we explore a unique scenario with newly introduced scales called degrees Elvis (E) and degrees Madonna (M), which are hypothetically linearly related. As with more familiar scales like Celsius and Fahrenheit, the key is to find a consistent formula to convert between them.Temperature conversion between scales typically involves:

• Identifying at least two equivalent measurements for calibration.
• Using these reference points to determine the relationship (which often takes the form of a linear equation).

Here, we were given these calibration points:

• \(40^{\circ} E\) equals \(25^{\circ} M\).
• \(280^{\circ} E\) equals \(125^{\circ} M\).

These points help us express one temperature scale in terms of another. By finding the slope of their relationship, we can create a conversion formula. This same approach is used in standard scales, allowing consistent temperature readings across different systems. It's crucial that the conversion accurately reflects the relationship and provides clarity when switching scales.