Let x denote a temperature on the Celsius scale, and let y...

Let $x$ denote a temperature on the Celsius scale, and let $y$ denote the corresponding temperature on the Fahrenheit scale. (a) Find a linear function relating $x$ and $y,$ use the facts that $32^{\circ} \mathrm{F}$ corresponds to $0^{\circ} \mathrm{C}$ and $212^{\circ} \mathrm{F}$ corresponds. to $100^{\circ} \mathrm{C} .$ Write the function in the form $y=A x+B$ (b) What Celsius temperature corresponds to $98.6^{\circ} \mathrm{F}$ ? (c) Find a number $z$ for which $z^{\prime} \mathrm{F}=z^{\prime} \mathrm{C}$.

Let $x$ denote a temperature on the Celsius scale, and let $y$ denote the corresponding temperature on the Fahrenheit scale.
(a) Find a linear function relating $x$ and $y,$ use the facts that $32^{\circ} \mathrm{F}$ corresponds to $0^{\circ} \mathrm{C}$ and $212^{\circ} \mathrm{F}$ corresponds.
to $100^{\circ} \mathrm{C} .$ Write the function in the form $y=A x+B$
(b) What Celsius temperature corresponds to $98.6^{\circ} \mathrm{F}$ ?
(c) Find a number $z$ for which $z^{\prime} \mathrm{F}=z^{\prime} \mathrm{C}$.

Key Concepts

Fixed Point in Temperature Conversion

A fixed point in the context of temperature conversion is a unique temperature value that is the same on both scales, meaning the numerical value in Celsius equals the numerical value in Fahrenheit. Identifying this point involves setting the conversion equation equal to its inverse, which provides insight into the intrinsic relationship between the two temperature scales.

Temperature Conversion

Temperature conversion involves converting a temperature measurement from one scale to another using a defined relationship. For Celsius and Fahrenheit, the conversion is linear and is based on specific fixed reference points, such as the freezing and boiling points of water. The standard conversion formula is derived from these fixed points and allows for an accurate transformation of temperature values between scales.

Linear Function

A linear function is a function whose graph is a straight line. It is generally expressed in the form y = mx + b, where m represents the slope of the line and b its y-intercept. In temperature conversion, the relationship between two scales is modeled as a linear function, because the change in one unit on one scale results in a constant change on the other scale.

Slope-Intercept Form

The slope-intercept form of a linear equation, y = mx + b, is a common way to represent a line. The slope, m, indicates the rate of change between the independent and dependent variables, while the y-intercept, b, indicates where the line crosses the y-axis. This form is particularly useful in converting temperatures between Celsius and Fahrenheit, where m and b are determined by known conversion points.

Transcript

00:01 So since we know we're using x and y, x is celsius, and y is fahrenheit, we will have 032 as a point for the freezing point, and 100 ,212 as the boiling point.

00:23 And what this tells us now is we can find slope, and we want to put it in the form y equals ax plus b or mx plus b.

00:32 So the slope of this will be 212 minus 32 over 100 minus 0.

00:45 So 212 minus 32 will be 180.

00:51 That's going to be over 100.

00:53 So that will end up giving us be 9 over 5 if we divide it by 20.

01:02 So 9 5ths is our slope.

01:08 And then we need to get the b value.

01:11 So we'll use this point here.

01:12 So we'll have 32 equals 9 5th times 0 plus b.

01:22 This will equal 0, so b equals 32, which means that we have our equation now in the form, y equals 9 5ths x plus, 32.

01:37 So that right there is our equation.

01:39 And then for part b, we're being asked to say what celsius temperature corresponds to 98 .6 degrees, the body temperature.

01:51 So since celsius is our x coordinate, we are going to put fahrenheit as our y coordinate.

02:00 So 98 .6 degrees equals 9 5ths x plus, plus 32.

02:08 So in order to determine which celsius temperature corresponds, we are going to solve for x.

02:13 So we will first start by subtracting 32.

02:17 So 98 .6 minus 32 is going to be 66 .6 .6 .6 .6 .6 .6...

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