Question

1. problem solving while on vacation in canada, you notice that temperatures are reported in degrees celsius. you know there is a linear relationship between fahrenheit and celsius, but you forget the formula. from science class, you remember the freezing point of water is 0°c or 32°f, and its boiling point is 100°c or 212°f. a. write an equation that represents degrees fahrenheit in terms of degrees celsius. b. the temperature outside is 22°c. what is this temperature in degrees fahrenheit? c. rewrite your equation in part (a) to represent degrees celsius in terms of degrees fahrenheit. d. the temperature of the hotel pool water is 83°f. what is this temperature in degrees celsius?

Explanation:

Step1: Find the slope of the linear - relationship

The two points on the linear relationship are $(0,32)$ (where $C = 0$ and $F=32$) and $(100,212)$ (where $C = 100$ and $F = 212$). The slope $m$ of the line $y=mx + b$ (where $y$ is $F$ and $x$ is $C$) is $m=\frac{212 - 32}{100-0}=\frac{180}{100}=\frac{9}{5}$. The $y$-intercept $b = 32$. So the equation for $F$ in terms of $C$ is $F=\frac{9}{5}C+32$.

Step2: Find $F$ when $C = 22$

Substitute $C = 22$ into the equation $F=\frac{9}{5}C+32$. Then $F=\frac{9}{5}\times22 + 32=\frac{198}{5}+32=39.6+32 = 71.6$.

Step3: Rewrite the equation for $C$ in terms of $F$

Start with $F=\frac{9}{5}C+32$. First, subtract 32 from both sides: $F - 32=\frac{9}{5}C$. Then multiply both sides by $\frac{5}{9}$ to get $C=\frac{5}{9}(F - 32)$.

Step4: Find $C$ when $F = 83$

Substitute $F = 83$ into the equation $C=\frac{5}{9}(F - 32)$. Then $C=\frac{5}{9}(83 - 32)=\frac{5}{9}\times51=\frac{255}{9}\approx28.3$.

Answer:

a. $F=\frac{9}{5}C+32$
b. $71.6^{\circ}F$
c. $C=\frac{5}{9}(F - 32)$
d. $\frac{255}{9}\approx28.3^{\circ}C$