Question

The relationship between the Fahrenheit (F) and Celsius (C) temperature scales is given by the equation$F=\frac{9}{5}C+32.$

(a)Complete the table to compare the two scales at the given values.

(b)Find the temperature at which the scales agree.

[Hint:Suppose that is the temperature at which the scales agree. Set$F=a$ and $C=a$. Then solve fora.]

Short answer

(a)

(b) The scales agrees at$-{40}^{\circ }C.$

Step-by-step solution

Part a. Step 1. Concept of Celsius and Fahrenheit.

Celsius scale is a temperature scale that is based on the freezing point of water at $0^{\circ }C$and the boiling point of water at ${100}^{\circ }C.$Fahrenheit scale is a temperature scale that is based on the freezing point at ${32}^{\circ }F$ and the boiling point at${212}^{\circ }F.$

Part a. Step 2. Determine the  and  for the given value.

The given equation is,

$F=\frac{9}{5}\left(C\right)+32$

When $C=-{30}^{\circ }C,$

$\begin{array}{l}F=\frac{9}{5}\left(-30\right)+32\\ F=-54+32\\ F=-{22}^{\circ }F\end{array}$

When $C=-{20}^{\circ }C,$

$\begin{array}{l}F=\frac{9}{5}\left(-20\right)+32\\ F=-36+32\\ F=-4^{\circ }F\end{array}$

When $C=-{10}^{\circ }C,$

$\begin{array}{l}F=\frac{9}{5}\left(-10\right)+32\\ F=-18+32\\ F={14}^{\circ }F\end{array}$

When $C=0^{\circ }C,$

$\begin{array}{l}F=\frac{9}{5}\left(0\right)+32\\ F={32}^{\circ }F\end{array}$

When $F={50}^{\circ }F,$

$\begin{array}{l}50=\frac{9}{5}\left(C\right)+32\\ C=\left(\frac{5}{9}\right)\left(50-32\right)\\ C={10}^{\circ }C\end{array}$

When $F={68}^{\circ }F,$

$\begin{array}{l}68=\frac{9}{5}\left(C\right)+32\\ C=\left(\frac{5}{9}\right)\left(68-32\right)\\ C={20}^{\circ }C\end{array}$

When $F={86}^{\circ }F,$

$\begin{array}{l}86=\frac{9}{5}\left(C\right)+32\\ C=\left(\frac{5}{9}\right)\left(86-32\right)\\ C={30}^{\circ }C\end{array}$

Part a. Step 3. Complete the table.

${}^{\circ }C$	${}^{\circ }F$
$-30$	$-22$
$-20$	$-4$
$-10$	14
0	32
10	50
20	68
30	86

Part b. Step 1. Concept of Celsius and Fahrenheit.

Celsius scale is a temperature scale that is based on the freezing point of water at $0^{\circ }C$and the boiling point of water at ${100}^{\circ }C.$Fahrenheit scale is a temperature scale that is based on the freezing point at ${32}^{\circ }F$ and the boiling point at${212}^{\circ }F.$

Part b. Step 2. Calculation of scale agree point.

Let the temperature at which the scale agrees be $a^{\circ }.$Substituting$a^{\circ }$ in the equation, we have to solve equation for $a^{\circ }.$

Part b. Step 3. Evaluating the temperature at which the scales agree.

Let the temperature at which the scale agrees be $a^{\circ }.$

Substituting $F=a^{\circ },C=a^{\circ },$in the equation, we get

$\begin{array}{c}a=\frac{9}{5}\left(a\right)+32\\ 5a=9a+160\\ -4a=160\\ a=-{40}^{\circ }C\end{array}$

Hence, The scales agrees at$-{40}^{\circ }C.$