Chapter 5.2, Problem 90AYU

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.

Height and Head Circumference The head circumference $C$ of a child is related to the height $H$ of the child (both in inches) through the function

$H\left(C\right)=2.15C-10.53$

(a) Express the head circumference $C$ as a function of height $H$.

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

(c) Predict the head circumference of a child who is 26 inches tall.

To determine

To find: The head circumference $C$ of a child is related to the height $H$ of the child (both in inches) through the function.

$H\left(C\right)=2.15C-10.53$

a. Express the head circumference $C$ as a function of height $H$.

Answer to Problem 90AYU

Solution:

a. The head circumference $C$ as a function of height $H$ is

$C=C\left(H\right)=\frac{H+10.53}{2.15}$

Explanation of Solution

Given:

The height function $H\left(C\right)=2.15C-10.53$.

Calculation:

a. We can find the head circumference $C$ in terms of $H$ as

$H\hspace{0.17em}=\hspace{0.17em}2.15C\hspace{0.17em}-\hspace{0.17em}10.53$

$\Rightarrow H+10.53=2.15C$

$\Rightarrow 2.15C=H+10.53$

$\Rightarrow C=C\left(H\right)=\frac{H+10.53}{2.15}$

To determine

To find: The head circumference $C$ of a child is related to the height $H$ of the child (both in inches) through the function.

$H\left(C\right)=2.15C-10.53$

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

Answer to Problem 90AYU

Solution:

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

Explanation of Solution

Given:

The height function $H\left(C\right)=2.15C-10.53$.

Calculation:

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

$\Rightarrow C\left(H\right)=C\left(H\left(C\right)\right)=C\left(2.15C-10.53\right)$

$\Rightarrow C\left(H\right)=\frac{2.15C-10.53+10.53}{2.15}$

$\Rightarrow C\left(H\right)=\frac{2.15C}{2.15}$

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

Similarly, we can prove

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

$\Rightarrow H\left(C\right)=H\left(\frac{H+10.53}{2.15}\right)$

$\Rightarrow H\left(C\right)=2.15\left(\frac{H+10.53}{2.15}\right)-10.53$

$\Rightarrow H\left(C\right)=H+10.53-10.53$

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

To determine

To find: The head circumference $C$ of a child is related to the height $H$ of the child (both in inches) through the function.

$H\left(C\right)=2.15C-10.53$

c. Predict the head circumference of a child who is 26 inches tall.

Answer to Problem 90AYU

Solution:

c. $C=16.9907=17 \text{inches}$.

Explanation of Solution

Given:

The height function $H\left(C\right)=2.15C-10.53$.

Calculation:

c. The head circumference of a child who is 26 inches tall.

We know that $C=C\left(H\right)=\frac{H+10.53}{2.15}$.

Given $H=26 \text{inches}$.

Therefore, $C=\frac{26+10.53}{2.15}=\frac{36.53}{2.15}$.

$C=16.9907=17 \text{inches}$.