Chapter 5.2, Problem 92AYU

In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using $y=f\left(x\right)$ to represent a function, an applied problem might use $C=C\left(q\right)$ to represent the cost $C$ of manufacturing q units of a good. Because of this, the inverse notation $f^{-1}$ used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as $C=C\left(q\right)$ will be $q=q\left(C\right)$. So $C=C\left(q\right)$ is a function that represents the cost $C$ as a function of the number $q$ of units manufactured, and $q=q\left(C\right)$ is a function that represents the number $q$ as a function of the cost $C$. Problems 91-94 illustrate this idea.

Temperature Conversion The function $F\left(C\right)=\frac{9}{5}C+32$ converts a temperature from $C$ degrees Celsius to $F$ degrees Fahrenheit.

(a) Express the temperature in degrees Celsius $C$ as a function of the temperature in degrees Fahrenheit $F$.

(b) Verify that $C=C\left(F\right)$ is the inverse of $F=F\left(C\right)$ by showing that $C\left(F\left(C\right)\right)=C$ and $F\left(C\left(F\right)\right)=F$.

(c) What is the temperature in degrees Celsius if it is 70 degrees Fahrenheit?

To determine

To find: The function $F\left(c\right)=\frac{9}{5}C+32$ converts a temperature from $C$ degrees Celsius to $F$ degrees Fahrenheit.

a. Express the temperature in degrees Celsius $C$ as a function of the temperature in degrees Fahrenheit $F$.

Answer to Problem 92AYU

Solution:

a. The temperature in degrees Celsius $C$ as a function of the temperature in degrees Fahrenheit $F$ as $C=\frac{5\left(F-32\right)}{9}$.

Explanation of Solution

Given:

The function $F$ converts a temperature from $C$ degrees Celsius to $F$ degrees Fahrenheit as $F\left(c\right)=\frac{9}{5}C+32$.

Calculation:

a. We can express temperature in degrees Celsius $C$ as a function of the temperature in degrees Fahrenheit $F$.

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

$\Rightarrow F= \frac{9}{5}C+32$

$\Rightarrow F-32= \frac{9}{5}C$

$\Rightarrow F-32= \frac{9}{5}C$

$\Rightarrow 5\left(F-32\right)= 9C$

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

To determine

To find: The function $F\left(c\right)=\frac{9}{5}C+32$ converts a temperature from $C$ degrees Celsius to $F$ degrees Fahrenheit.

b. Verify that $C=C\left(F\right)$ is the inverse of $F=F\left(C\right)$ by showing that $C\left(F\left(C\right)\right)=C$ and $F\left(C\left(F\right)\right)=F$.

Answer to Problem 92AYU

Solution:

b. Verified that $C=C\left(F\right)$ is the inverse of $F=F\left(C\right)$.

Explanation of Solution

Given:

The function $F$ converts a temperature from $C$ degrees Celsius to $F$ degrees Fahrenheit as $F\left(c\right)=\frac{9}{5}C+32$.

Calculation:

b. Verify $C=C\left(F\right)$ is the inverse of $F=F\left(C\right)$ by showing below.

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

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

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

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

$\Rightarrow C\left(F\right)=C$

Similarly, we can prove

$\Rightarrow F\left(C\right)=F\left(C\left(F\right)\right)$

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

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

$\Rightarrow F\left(C\right)=\left(\frac{9C}{9}\right)$

$\Rightarrow F\left(C\right)=C$

To determine

To find: The function $F\left(c\right)=\frac{9}{5}C+32$ converts a temperature from $C$ degrees Celsius to $F$ degrees Fahrenheit.

c. What is the temperature in degrees Celsius if it is 70 degrees Fahrenheit?

Answer to Problem 92AYU

Solution:

c. The temperature in degrees Celsius if it is 70 degrees Fahrenheit as.

$\text{C}=21\text{ degrees Celsius}$.

Explanation of Solution

Given:

The function $F$ converts a temperature from $C$ degrees Celsius to $F$ degrees Fahrenheit as $F\left(c\right)=\frac{9}{5}C+32$.

Calculation:

c. The temperature in degrees Celsius if it is 70 degrees Fahrenheit as

We know that $C = \frac{5\left(F - 32\right)}{9}$.

$\Rightarrow$Given $F=70\text{ degrees}$.

Therefore, $C=\frac{5\left(70-32\right)}{9}$.

$C=\frac{5\left(38\right)}{9}$

$C=\frac{190}{9}$

$C=21.1111 \text{Celsius}$.

$C=21 \text{degrees }\text{Celsius}$.