Q48. - Geometry

Question

The ruler postulate suggests that there are many ways to assign coordinates to a line. The Fahrenheit and Celsius temperature scales on a thermometer indicate two such ways of assigning coordinates. A Fahrenheit temperature of $\mathbf{32}\mathbf{°}$ corresponds to a Celsius temperature of ${\mathbf{0}}^{\mathbf{\circ }}$. The formula, or rule for converting a Fahrenheit temperature F into a Celsius temperature C is $\mathit{C}\mathbf{=}\frac{\mathbf{5}}{\mathbf{9}}\left(F-32\right)$.

a. What Celsius temperatures corresponds to Fahrenheit temperature of ${\mathbf{212}}^{\mathbf{\circ }}\mathbf{,}\mathbf{98}{\mathbf{.6}}^{\mathbf{\circ }}$?

b. Solve the equation above for F to obtain a rule for converting Celsius temperature to Fahrenheit temperatures.

c. What Fahrenheit temperatures correspond to Celsius temperatures of $\mathbf{-}{\mathbf{40}}^{\mathbf{\circ }}\mathbf{,}{\mathbf{2000}}^{\mathbf{\circ }}$?

Step-by-Step Solution

Answer

a. In Celsius $\mathit{C}\mathbf{=}{\mathbf{100}}^{\mathbf{\circ }}$ and $\mathit{C}\mathbf{=}{\mathbf{37}}^{\mathbf{\circ }}$ respectively.

b. The equation to convert the Celsius to Fahrenheit temperature is $\mathit{F}\mathbf{=}\frac{\mathbf{9}\mathbf{C}\mathbf{+}\mathbf{160}}{\mathbf{5}}$.

c. The Fahrenheit temperature is $\mathit{F}\mathbf{=}\mathbf{-}\mathbf{40}\mathbf{°}$ and $\mathit{F}\mathbf{=}\mathbf{3632}\mathbf{°}$ respectively.

a. Step 1- Check the figure.

Consider the figure.

a. Step 2- Apply the concept of ruler postulate.

The Ruler Postulate suggests that there are many ways to assign coordinates to a line. The Fahrenheit and Celsius temperature scales on a thermometer indicate two such ways of assigning coordinates. A Fahrenheit temperature of $\mathbf{32}\mathbf{°}$ corresponds to a Celsius temperature of $\mathbf{0}\mathbf{°}$.

a. Step 3- Step description.

A Fahrenheit temperature of $32°$ corresponds to Celsius temperature or $0°$. The formula for converting a Fahrenheit temperature F into Celsius temperature C is $C=\frac{5}{9}\left(F-32\right)$.

Substitute value of $F={212}^{\circ }\text{\hspace{0.17em}in\hspace{0.17em}}C=\frac{5}{9}\left(F-32\right)$.

$\begin{array}{l}C=\frac{5}{9}\left(212-32\right)\\ =\frac{5\times 180}{9}\\ =5\times 20\\ C=100\end{array}$ 

Substitute value of $F={98.6}^{\circ }\text{\hspace{0.17em}in\hspace{0.17em}}C=\frac{5}{9}\left(F-32\right)$

$\begin{array}{l}C=\frac{5}{9}\left(98.6-32\right)\\ =\frac{5\times 66.6}{9}\\ =5\times 7.4\\ C=37\end{array}$

 

Hence, in Celsius, the temperature is $C=100°$ and $C=37°$ respectively.

b. Step 1- Check the figure.

Consider the figure.

b. Step 2- Apply the concept of ruler postulate.

The Ruler Postulate suggests that there are many ways to assign coordinates to a line. The Fahrenheit and Celsius temperature scales on a thermometer indicate two such ways of assigning coordinates. A Fahrenheit temperature of $\mathbf{32}\mathbf{°}$ corresponds to a Celsius temperature of $\mathbf{0}\mathbf{°}$.

b. Step 3- Step description.

A Fahrenheit temperature of $32°$ corresponds to Celsius temperature or $0°$. The formula for converting a Fahrenheit temperature F into Celsius temperature C is $C=\frac{5}{9}\left(F-32\right)$

From the equation

$\begin{array}{l}C=\frac{5}{9}\left(F-32\right)\\ 9C=5\left(F-32\right)\\ 9C=5F-160\\ 5F=9C+160\end{array}$

Solve the obtained equation further as follows:

$F=\frac{9C+160}{5}$

Hence, the value of the above expression is $F=\frac{9C+160}{5}$.

c. Step 1- Check the figure.

Consider the figure.

c. Step 2- Apply the concept of ruler postulate.

The Ruler Postulate suggests that there are many ways to assign coordinates to a line. The Fahrenheit and Celsius temperature scales on a thermometer indicate two such ways of assigning coordinates. A Fahrenheit temperature of $\mathbf{32}\mathbf{°}$ corresponds to a Celsius temperature of $\mathbf{0}\mathbf{°}$.

c. Step 3- Step description.

A Fahrenheit temperature of $32°$ corresponds to Celsius temperature or $0°$. The formula for converting a Fahrenheit temperature F into Celsius temperature C is $C=\frac{5}{9}\left(F-32\right)$

From part (b) the equation is $F=\frac{9C+160}{5}$. 

Substitute value of $C=-{40}^{\circ }$ in the equation $F=\frac{9C+160}{5}$.

$\begin{array}{l}F=\frac{9\times \left(-40\right)+160}{5}\\ =\frac{-360+160}{5}\\ =\frac{-200}{5}\\ F=-40\end{array}$

Substitute value of $C={2000}^{\circ }$ in the equation $F=\frac{9C+160}{5}$ as follows:

$\begin{array}{l}F=\frac{9\times \left(2000\right)+160}{5}\\ =\frac{18000+160}{5}\\ =\frac{18160}{5}\\ F=3632\end{array}$

Hence, the Fahrenheit temperature is $F=-40°$ and $F=3632°$ respectively.