Chapter 5.3, Problem 6MRPS

Solution

a.

To calculate: Write an equation that relates $b,w$ and $h$ .

The equation $b,w$ relating  $h$ and is thus: $b=\frac{w}{h^2}$ , where  $w$  is in kilograms and  $h$  is in meters.

Given information:

A person who is $1.6m$ tall,

Weight = $51.2kg$

Body mass index = $20$

Concept used:

Body mass index rule; $b=c\frac{w}{h^2}$

Calculation:

Given that: $b=c\frac{w}{h^2}$

Where $c$ is constant;

Plug the given information:

  $\begin{array}{l}20=c\frac{51.2kg}{{\left(1.6m\right)}^2}\text{}\\ c=1\frac{m^2}{kg}\end{array}$

Conclusion:

  $b=\frac{w}{h^2}$

b.

To calculate: Complete the given table.

$b$	$h$	$w$
$20$	$\sqrt{\frac{45}{20}}=1.5$	$45$
$\frac{41}{{1.4}^2}\approx 20.92$	$1.4$	$41$
$19$	$1.5$	$19\times {\left(1.5\right)}^2=42.75$

Given information:

A person who is $1.6m$ tall,

Weight = $51.2kg$

Body mass index = $20$

Concept used:

Body Mass Index Rule

Calculation:

We are given the function:  $b=\frac{w}{h^2}$

In order to complete the table, let's solve the equation for each variable.

Solving for  $w$ : $w=b{h}^2$

Solving for  $h$ :

  $\begin{array}{l}b{h}^2=w\\ h^2=\frac{w}{b}\\ h=\sqrt{\frac{w}{b}}\end{array}$

Now complete the table:

$b$	$h$	$w$
$20$	$\sqrt{\frac{45}{20}}=1.5$	$45$
$\frac{41}{{1.4}^2}\approx 20.92$	$1.4$	$41$
$19$	$1.5$	$19\times {\left(1.5\right)}^2=42.75$

c.

To calculate: Compare the body mass indices of two people.

Taller person's BMI is about  $\text{82}.\text{6}\%$ that of the shorter person's.

Given information:

The height of the one person is $10\%$ greater than the height of other person.

Concept used:

Body Mass Index Rule

Calculation:

The body mass index  ? $b_1$  for someone of height  $h$ and weight is $w$

  $b_1=\frac{w}{h^2}$

For another person who has the same weight but is $10\%$ taller,

  $(1+0.10)h=1.1h$

Into the formula:

  $b_2=\frac{w}{{\left(1.1h\right)}^2}$

  $\begin{array}{l}\\ b_2=\frac{1}{1.21}\frac{w}{{\left(h\right)}^2}\\ b_2=\frac{b_1}{1.21}\\ \\ b_2\approx 0.826b_1\end{array}$

Conclusion:

Taller person's BMI is about  $\text{82}.\text{6}\%$ that of the shorter person's.