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Chapter 1: Problem 99

Solve each problem. Celsius to Falirenheit Formula Fahrenheit temperature \(F\) is a linear function of Celsius temperature \(C\). The ordered pair \((0,32)\) is an ordered pair of this function because \(0^{\circ} \mathrm{C}\) is equivalent to \(32^{\circ} \mathrm{F},\) the freezing point of water. The ordered pair \((100,212)\) is also an ordered pair of this function because \(100^{\circ} \mathrm{C}\) is equivalent to \(212^{\circ} \mathrm{F},\) the boiling point of water. Use the two given points and the point-slope formula to write \(F\) as a function of \(C .\) Find the Fahrenheit temperature of an oven at \(150^{\circ} \mathrm{C}\)

Short Answer

Expert verified
The function is \[ F = 1.8C + 32 \], and \( F = 302 \) at \(150^{\box} C \).

Step by step solution

01

Identify the given points

The given points are \(0, 32\) and \(100, 212\).These points represent Celsius and Fahrenheit temperatures: \(C = 0, F = 32\) and \(C = 100, F = 212\).
02

Calculate the slope

Use the formula for the slope: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Where \(x_1 = 0, y_1 = 32\) and \(x_2 = 100, y_2 = 212\). \[ m = \frac{212 - 32}{100 - 0} = \frac{180}{100} = 1.8 \].
03

Use point-slope formula

The point-slope formula for a line is \[ y - y_1 = m(x - x_1) \]. With the point \(0, 32\) and slope \(m = 1.8\), substitute these values into the formula: \[ F - 32 = 1.8(C - 0) \]. Simplify to obtain the function: \[ F = 1.8C + 32 \].
04

Find Fahrenheit temperature at 150掳C

To determine the Fahrenheit temperature when \(C = 150\), substitute 150 in place of \(C\) in the function \[ F = 1.8C + 32 \]: \[ F = 1.8(150) + 32 = 270 + 32 = 302 \].

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Key Concepts

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

linear functions
A linear function describes a straight line in a coordinate plane. These functions can be written in the form of y = mx + b, where:
  • y represents the dependent variable.
  • x is the independent variable.
  • m stands for the slope of the line.
  • b is the y-intercept, where the line crosses the y-axis.
In our temperature conversion problem, the function expresses the relationship between Celsius ( C) and Fahrenheit ( F) temperatures. The slope ( m) denotes how much F increases for every increase in C. In this case, the equation F = 1.8C + 32 outlines a clear linear relationship where 1.8 is the slope and 32 is the y-intercept, representing the freezing point of water. Understanding linear functions helps break down how changes in one variable affect the other. It simplifies predictions and calculations across various applications.
point-slope formula
The point-slope formula is a method used to find the equation of a line when you know one point on the line and the slope. The formula is expressed as: y - y_1 = m(x - x_1), where:
  • (x_1, y_1) represents a known point on the line.
  • m is the slope.
To illustrate how this works, let鈥檚 look at our problem. We were given two points: (0, 32) and (100, 212). We first found the slope ( m) using the formula m = (y_2 - y_1) / (x_2 - x_1). Substituting these into the point-slope formula used the point (0, 32) and solved for F, the Fahrenheit temperature. This process provided us with our final function: F = 1.8C + 32. Point-slope is invaluable for quickly finding linear relationships.
temperature conversion
Converting temperature between Celsius and Fahrenheit is a common practical use of linear equations. The relationship between Celsius ( C) and Fahrenheit ( F) temperatures can be expressed with the linear function: F = 1.8C + 32.
Here鈥檚 a breakdown of how to use this equation:
  • To find Fahrenheit when you have Celsius, multiply C by 1.8 and add 32.
  • For Celsius when given Fahrenheit, you can rearrange the formula: C = (F - 32) / 1.8.
For example, to find Fahrenheit at 150掳C, substitute 150 for C in the formula, so F = 1.8(150) + 32 = 302掳F. This temperature conversion technique is essential in various scientific, engineering, and day-to-day contexts, helping us understand and compare temperature scales.

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Most popular questions from this chapter

Carpenters and Helpers Because a job was finished early, a contractor will distribute a total of \(\$ 2400\) in bonuses to 9 carpenters and 3 helpers. The carpenters all get the same amount and the helpers all get the same amount. Write the amount of a helper's bonus as a function of the amount of a carpenter's bonus. Find and interpret the slope.

Recall that \(\sqrt{w}\) is a real number only if \(w \geq 0\) and \(1 / w\) is a real mumber only if \(w \neq 0 .\) For what values of \(x\) is each of the following expressions a real mumber? $$\frac{1}{\sqrt{2-x}}$$

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Equal Slopes Show that if \(y=m x+b_{1}\) and \(y=m x+b_{2}\) are equations of lines with equal slopes, but \(b_{1} \neq b_{2}\), then they have no point in common. Assume that they have a point in common and see that this assumption leads to a contradiction.
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