Chapter 5.2, Problem 91AYU

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.

Ideal Body Weight One model for the ideal body weight $W$ for men (in kilograms) as a function of height $h$ (in inches) is given by the function

$W\left(h\right)=50+2.3\left(h-60\right)$

(a) What is the ideal weight of a 6-foot male?

(b) Express the height $h$ as a function of weight $W$.

(c) Verify that $h=h\left(W\right)$ is the inverse of $W=W\left(h\right)$ by showing that $h\left(W\left(h\right)\right)=h$ and $W\left(h\left(W\right)\right)=W$.

(d) What is the height of a male who is at his ideal weight of 80 kilograms?

[Note: The ideal body weight $W$ for women (in kilograms) as a function of height $h$ (in inches) is given by $W\left(h\right)=45.5+2.3\left(h-60\right)$.

To determine

To find: One model for the ideal body weight $W$ for men (in kilograms) as a function of height $h$ (in inches) is given by the function.

$w\left(h\right)=50+2.3\left(h-60\right)$

a. What is the ideal weight of a 6-foot male?

Answer to Problem 91AYU

Solution:

For Men: $W\left(h\right)=50+2.3\left(h-60\right)$

a. The ideal weight of a 6 – foot male is $W = 77.6 \text{kilos}$.

For Women: $W\left(h\right)=45.5+2.3\left(h-60\right)$

a. The ideal weight of a 6 – foot female is $W\hspace{0.17em}=\hspace{0.17em}73.1 \text{kilos}$.

Explanation of Solution

Given:

The ideal body weight $W$ for men as a height function.

$w\left(h\right)=50+2.3\left(h-60\right)$

Calculation:

a. Given the ideal weight of a 6-foot male.

i.e., $h=6 \text{foot}=72 \text{inches}$

$W=50+2.3\left(72-60\right)$

$W=50+2.3\left(12\right)$

$W=50+27.6$

$W=77.6 \text{kilos}$

To determine

To find: One model for the ideal body weight $W$ for men (in kilograms) as a function of height $h$ (in inches) is given by the function.

$w\left(h\right)=50+2.3\left(h-60\right)$

b. Express the height $h$ as a function of weight $W$.

Answer to Problem 91AYU

Solution:

For Men: $W\left(h\right)=50+2.3\left(h-60\right)$

b. The height $h$ as a function of weight $W,h=\frac{W+88}{2.3}$.

For Women: $W\left(h\right)=45.5+2.3\left(h-60\right)$

b. The height $h$ as a function of weight $W,h=\frac{W+92.5}{2.3}$.

Explanation of Solution

Given:

The ideal body weight $W$ for men as a height function.

$w\left(h\right)=50+2.3\left(h-60\right)$

Calculation:

b. We can express $h$ as a function of weight $W$.

Given $W\left(h\right)\hspace{0.17em}=\hspace{0.17em}50\hspace{0.17em}+\hspace{0.17em}2.3\left(h\hspace{0.17em}-\hspace{0.17em}60\right)$.

$\Rightarrow W-50=2.3\left(h-60\right)$

$\Rightarrow h-60=\frac{W-50}{2.3}$

$\Rightarrow h=\frac{W-50}{2.3}+60$

$\Rightarrow h=\frac{W-50+138}{2.3}$

$\Rightarrow h=\frac{W+88}{2.3}$

To determine

To find: One model for the ideal body weight $W$ for men (in kilograms) as a function of height $h$ (in inches) is given by the function.

$w\left(h\right)=50+2.3\left(h-60\right)$

c. Verify that $h=h\left(W\right)$ is the inverse of $W=W\left(h\right)$ by showing that $h\left(W\left(h\right)\right)=h$ and $W\left(h\left(w\right)\right)=W$.

Answer to Problem 91AYU

Solution:

For Men: $W\left(h\right)=50+2.3\left(h-60\right)$

c. Verified that $h=h\left(W\right)$ is the inverse of $W=W\left(h\right)$.

For Women: $W\left(h\right)=45.5+2.3\left(h-60\right)$

c. Verified that $h=h\left(W\right)$ is the inverse of $W=W\left(h\right)$.

Explanation of Solution

Given:

The ideal body weight $W$ for men as a height function.

$w\left(h\right)=50+2.3\left(h-60\right)$

Calculation:

c. Verify $h=h\left(W\right)$ is the inverse of $W=W\left(h\right)$ by showing below.

$\Rightarrow h\left(W\left(h\right)\right)=h\left(W\left(h\right)\right)=h\left(50+2.3\left(h-60\right)\right)$

$\Rightarrow h\left(W\left(h\right)\right)=\frac{50+2.3\left(h-60\right)+88}{2.3}$

$\Rightarrow h\left(W\left(h\right)\right)=\frac{138+2.3\left(h\right)-138}{2.3}$

$\Rightarrow h\left(W\left(h\right)\right)=\frac{2.3\left(h\right)}{2.3}$

$\Rightarrow h\left(W\left(h\right)\right)=h$

Similarly, we can prove

$\Rightarrow W\left(h\right)=W\left(h\left(W\right)\right)$

$\Rightarrow W\left(h\right)=W\left(\frac{W+88}{2.3}\right)$

$\Rightarrow W\left(h\right)=50+2.3\left(\frac{W+88}{2.3}-60\right)$

$\Rightarrow W\left(h\right)=50+W+88-138$

$\Rightarrow W\left(h\right)=138+W-138$

$\Rightarrow W\left(h\right)=W$

To determine

To find: One model for the ideal body weight $W$ for men (in kilograms) as a function of height $h$ (in inches) is given by the function.

$w\left(h\right)=50+2.3\left(h-60\right)$

d. What is the height of a male who is at his ideal weight of 80 kilograms? [Note: The ideal body weight $W$ for women (in kilograms) as a function of height $h$ (in inches) is given by $W\left(h\right)=45.5+2.31h-602$].

Answer to Problem 91AYU

Solution:

For Men: $W\left(h\right)=50+2.3\left(h-60\right)$

d. $h=73 \text{inches}=6.1 \text{feets}$.

For Women: $W\left(h\right)=45.5+2.3\left(h-60\right)$

d. $h=75\text{ inches}$.

Explanation of Solution

Given:

The ideal body weight $W$ for men as a height function.

$w\left(h\right)=50+2.3\left(h-60\right)$

Calculation:

d. The height of a male who is at his ideal weight of 80 kilograms as.

We know that $h=\frac{W-50}{2.3}+60$.

$\Rightarrow$ Given $W=80$ kilograms.

Therefore, $h=\frac{W-50}{2.3}+60=\frac{80-50}{2.3}+60$.

$h=\frac{30}{2.3}+60$

$h=\frac{30}{2.3}+60$

$h=13.04348+60$

$h=73.04348=73 \text{inches}$

$h=6.1 \text{feets}$.