Q. 91 - Precalculus Enhanced with Graphing Utilities

Question

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

(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  $W\left(h\right)=45.5+2.3(h-60)$.

Step-by-Step Solution

Answer

Part (a). The weight of $6$-foot male is $77.6 kg$.

Part (b). The expression of height as the function of weight is: $h\left(W\right)=\frac{W-50}{2.3}+60$.

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

Part (d). The height of male having weight $80kg$ is $73.04 inches$.

Part (a) Step 1. Find the weight

First converting $6$ foot into inches, we get:

$$\begin{gathered} 1 foot=12 inches \\ 6 foot= 12\times 6 inches \\ =72 inches \end{gathered}$$

Substituting $72$ in the equation $W\left(h\right)=50+2.3(h-60)$, we get:

$$\begin{gathered} W\left(h\right)=50+2.3(h-60) \\ W\left(72\right)=50+2.3\left(72-60\right) \\ W\left(72\right)=50+2.3\times 12 \\ W\left(72\right)=77.6 kg \end{gathered}$$

The weight of the male is $77.6kg$

Part (b) Step 1. Express in the terms of W

The given function is $W\left(h\right)=50+2.3\left(h-60\right)$.

Subtracting both sides by $50$, we get:

$$\begin{gathered} W-50=50+2.3\left(h-60\right)-50 \\ W-50=2.3\left(h-60\right) \end{gathered}$$

Dividing both sides by $2.3$, we get:

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

Adding both sides by $60$, we get:

$$\begin{gathered} \frac{W-50}{2.3}+60=h \\ h\left(W\right)=\frac{W-50}{2.3}+60 \end{gathered}$$

Part (c) Step 1. Verify the inverse

Substituting $W\left(h\right)$ in the function$h\left(W\right(h\left)\right)=h$, we get:

$$\begin{gathered} h(50+2.3(h-60\left)\right)=h \\ \frac{50+2.3(h-60)-50}{2.3}+60=h \\ \frac{2.3(h-60)}{2.3}+60=h \\ h-60+60=h \\ h=h \end{gathered}$$

This is true

Substituting $h\left(W\right)$ in the function$W\left(h\right(W\left)\right)=W$, we get:

$$\begin{gathered} W\left(\frac{W-50}{2.3}+60\right)=W \\ 50+2.3\left(\frac{W-50}{2.3}+60-60\right)=W \\ 50+W-50=W \\ W=W \end{gathered}$$

This is true.

Thus, it is verified that $h=h\left(W\right)$ is the inverse of $W=W\left(h\right)$. 

Part (d) Step 1. Find the height of male

Substituting $W=80$ in the equation $h\left(W\right)=\frac{W-50}{2.3}+60$, we get:

$$\begin{gathered} h\left(W\right)=\frac{W-50}{2.3}+60 \\ h\left(80\right)=\frac{80-50}{2.3}+60 \\ h\left(80\right)=\frac{30}{2.3}+60 \\ h\left(80\right)=73.04 inches \end{gathered}$$