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Chapter 1: Problem 61

GENERAL: Temperature On the Fahrenheit temperature scale, water freezes at \(32^{\circ}\) and boils at \(212^{\circ} .\) On the Celsius (centigrade) scale, water freezes at \(0^{\circ}\) and boils at \(100^{\circ}\). a. Use the two (Celsius, Fahrenheit) data points \((0,32)\) and \((100,212)\) to find the linear relationship \(y=m x+b\) between \(x=\) Celsius temperature and \(y=\) Fahrenheit temperature. b. Find the Fahrenheit temperature that corresponds to \(20^{\circ}\) Celsius.

Short Answer

Expert verified
The Fahrenheit temperature for 20掳C is 68掳F.

Step by step solution

01

Understand the Problem

We need to find the linear equation that relates Celsius and Fahrenheit using the given points (0,32) and (100,212), and then use that equation to determine the Fahrenheit equivalent of 20掳C.
02

Determine the Slope, m

The slope, \(m\), of a line is given by the change in \(y\) over the change in \(x\). Use the points \((x_1, y_1) = (0, 32)\) and \((x_2, y_2) = (100, 212)\).\\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8 \]
03

Find the Intercept, b

The equation of the line is \(y = mx + b\). Using one of the points, substitute to find \(b\). Use \((0, 32)\):\\[ 32 = 1.8 \times 0 + b \] \\[ b = 32 \]
04

Write the Linear Equation

Now that we have both \(m\) and \(b\), we can write the linear equation relating Fahrenheit \((y)\) and Celsius \((x)\):\\[ y = 1.8x + 32 \]
05

Convert 20掳C to Fahrenheit

Substitute \(x = 20\)掳C into the equation \(y = 1.8x + 32\) to find the corresponding Fahrenheit temperature:\\[ y = 1.8 \times 20 + 32 = 36 + 32 = 68 \]

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Key Concepts

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

Temperature Conversion
The relationship between Celsius and Fahrenheit scales is crucial to understand how different temperature systems work. Water, a common reference in temperature scales, freezes at 0掳C and boils at 100掳C. On the Fahrenheit scale, the same points are 32掳F and 212掳F, respectively. These differences in freezing and boiling points create a linear relationship between the two scales.
We can express this relationship as a linear equation where Celsius is represented by the variable \(x\) and Fahrenheit by \(y\), forming a line equation of the form \(y = mx + b\). Here, \(m\) is the slope indicating how many Fahrenheit degrees a Celsius degree corresponds to, and \(b\) is the y-intercept where the line crosses the y-axis.
The distinct freezing and boiling values provide us two points, namely \((0, 32)\) and \((100, 212)\), which are used to derive the linear equation. Understanding this conversion is immensely helpful in everyday life for weather readings, cooking, and scientific experiments.
Slope and Intercept
Intercept and slope are fundamental concepts in the mathematical representation of lines. The **slope**, \(m\), measures the steepness of the line, showing the rate of change between variables. In temperature conversion, this rate indicates how rapidly temperatures in Celsius convert to Fahrenheit. Calculated as \(\frac{y_2 - y_1}{x_2 - x_1}\), it becomes \(1.8\) from the points \((0, 32)\) to \((100, 212)\). This means for every 1掳C increase, Fahrenheit increases by 1.8掳F.
The **y-intercept**, \(b\), represents the point where the line crosses the y-axis. It indicates the Fahrenheit equivalent when Celsius is 0. In our temperature equation, it is \(32\), which means at 0掳C, it is 32掳F. By combining both, the equation \(y = 1.8x + 32\) is formed. This provides a clear method to transform temperatures between the scales.
Mathematical Modeling
Mathematical modeling involves representing real-world phenomena through equations or data sets. In the case of temperature conversion, we accurately represent the relationship between Celsius and Fahrenheit using a linear model. This model helps predict unknown values easily, like finding the Fahrenheit equivalent for a given Celsius temperature.
By deriving an equation from known data points, like the freezing and boiling points of water, we create a reliable model \(y = 1.8x + 32\). This model is valuable as it simplifies converting any Celsius reading into Fahrenheit confidently. For example, to find out what 20掳C equates to in Fahrenheit, simply substitute into the equation: \(y = 1.8 \times 20 + 32\), giving 68掳F.
Through mathematical modeling, complex real-world systems become understandable and predictable, aiding in planning and decision-making.

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Most popular questions from this chapter

ENVIRONMENTAL SCIENCE: Wind Energy The use of wind power is growing rapidly after a slow start, especially in Europe, where it is seen as an efficient and renewable source of energy. Global wind power generating capacity for the years 1996 to 2008 is given approximately by \(y=0.9 x^{2}-3.9 x+12.4\) thousand megawatts (MW), where \(x\) is the number of years after 1995 . (One megawatt would supply the electrical needs of approximately 100 homes). a. Graph this curve on the window \([0,20]\) by \([0,300]\). b. Use this curve to predict the global wind power generating capacity in the year \(2015 .\) [Hint: Which \(x\) -value corresponds to \(2015 ?\) Then use TRACE, EVALUATE, or TABLE.] c. Predict the global wind power generating capacity in the year \(2020 .\)

The intersection of an isocost line \(w L+r K=C\) and an isoquant curve \(K=a L^{b}\) (see pages 18 and 32 ) gives the amounts of labor \(L\) and capital \(K\) for fixed production and cost. Find the intersection point \((L, K)\) of each isocost and isoquant. [Hint: After substituting the second expression into the first, multiply through by \(L\) and factor.] $$ 5 L+4 K=120 \text { and } K=180 \cdot L^{-1} $$

65-66. BUSINESS: Isocost Lines An isocost line (iso means "same") shows the different combinations of labor and capital (the value of factory buildings, machinery, and so on) a company may buy for the same total cost. An isocost line has equation $$ w L+r K=\mathrm{C} \quad \text { for } L \geq 0, \quad K \geq 0 $$ where \(L\) is the units of labor costing \(w\) dollars per unit, \(K\) is the units of capital purchased at \(r\) dollars per unit, and \(C\) is the total cost. Since both \(L\) and \(K\) must be nonnegative, an isocost line is a line segment in just the first quadrant. a. Write the equation of the isocost line with \(w=10, \quad r=5, \quad C=1000\), and graph it in the first quadrant. b. Verify that the following \((L, K)\) pairs all have the same total cost. \((100,0), \quad(75,50), \quad(20,160), \quad(0,200)\)

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ \begin{array}{l} f(x)=x^{4} \\ \text { [Hint: Use }(x+h)^{4}=x^{4}+4 x^{3} h+6 x^{2} h^{2}+ \\ \left.4 x h^{3}+h^{4} .\right] \end{array} $$

True or False: \(x=3\) is a vertical line.
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