Chapter 1, Problem 113QP

Body mass index (BMI) is a number calculated from a person's weight and height. BMI provides a reliable indicator of body fatness for most people and is used to screen for weight categories that may lead to health problems. The formula for a person‘s BMI is

$\text{BMI}\text{=}\frac{\text{weight}\text{(kg)}}{{\text{[height}\text{(m)]}}^{\text{2}}}$

For an adult, a BMI value between 18.5 and 24.9 is healthy and considered normal.

(a) Calculate the BMI for a 6-ft, 2-in man who weighs 169 lb.

(b) Is this person underweight at a healthy weight, or overweight?

Interpretation Introduction

Interpretation:

The BMI of a man with a height of $6\text{ ft }2\text{ in}$ and a weight of $169\text{ lb}$ .

Explanation of Solution

The formula used to calculate BMI is as follows:

$\text{BMI}=\frac{\text{weight }}{{\left[\text{height}\right]}^2}$ …… $\left(1\right)$

The conversion of height $\left(6\text{ ft }2\text{ in}\right)$ to inch $\left(\text{in}\right)$ is as follows:

The relation between foot $\left(\text{ft}\right)$ and inch $\left(\text{in}\right)$ is shown below.

$1\text{ ft}=12\text{ in}$

The conversion factors used to convert $\text{ft}$ into $\text{in}$ are shown below.

$\frac{1\text{ ft}}{12\text{ in}}\text{or}\frac{12\text{ in}}{1\text{ ft}}$

$\left(6\text{ ft }2\text{ in}\right)$ is converted into $\text{in}$ using a conversion factor, as shown below.

$\begin{array}{c}\left(6\text{ ft }2\text{ in}\right)=6\text{ ft }2\text{ in}\\ \text{=}6\text{ ft}\times \frac{12\text{ in}}{1\text{ ft}}+2\text{ in}\\ \text{=72 in}+2\text{ in}\\ =74\text{ in}\end{array}$

Therefore, the height is calculated to be $74\text{ in}$ .

The conversion of the height $\left(\text{74 in}\right)$ into meters $\left(\text{m}\right)$ is:

The relation between $\text{in}$ and $\text{cm}$ is shown below.

$1\text{ in}=2.54\text{ cm}$

The conversion factors used to convert $\text{in}$ into $\text{cm}$ are shown below.

$\frac{1\text{ in}}{\text{2}\text{.54 cm}}\text{or}\frac{\text{2}\text{.54 cm}}{1\text{ in}}$

The relation between centimeter $\left(\text{cm}\right)$ and meter $\left(\text{m}\right)$ is shown below.

$1\text{ m}=100\text{ cm}$

The conversion factors used to convert $\text{cm}$ into $\text{m}$ are shown below.

$\frac{1\text{ m}}{100\text{ cm}}\text{or}\frac{100\text{ cm}}{1\text{ m}}$

$74\text{ in}$ is converted into $\text{m}$ using a conversion factor, as shown below.

$\begin{array}{c}74\text{ in}=74\text{ in}\times \frac{\text{2}\text{.54 cm}}{1\text{ in}}\times \frac{1\text{ m}}{100\text{ cm}}\\ =1.879\text{ m}\\ =1.88\text{ m}\end{array}$

The answer is rounded off by dropping the last number, which is more than $5$ , and the significant figure present on the left is odd. Therefore, the last significant figure will be increased by one unit. As a result, the final height (in meters), after rounding off,is $1.88\text{ m}$ .

The conversion of weight $\left(169\text{ lb}\right)$ into kilogram $\left(\text{kg}\right)$ is:

The relation between $\text{lb}$ and $\text{g}$ is shown below,

$1\text{ lb}=\text{453}\text{.6 g}$

The conversion factors used to convert $\text{lb}$ into $\text{g}$ are shown below.

$\frac{1\text{ g}}{\text{453}\text{.6 lb}}\text{or}\frac{\text{453}\text{.6 lb}}{1\text{ g}}$

The relation to convert $\text{g}$ into $\text{kg}$ is shown below.

$1\text{ kg}=\text{1000 g}$

The conversion factors used to convert $\text{g}$ into $\text{kg}$ are shown below.

$\frac{1\text{ kg}}{1000\text{ g}}\text{or}\frac{1000\text{ g}}{1\text{ kg}}$

$169\text{ lb}$ is converted into $\text{kg}$ using a conversion factor, as shown below.

$\begin{array}{c}169\text{ lb}=169\text{ lb}\times \frac{453.6\text{ g}}{1\text{ lb}}\times \frac{1\text{ kg}}{1000\text{ g}}\\ =76.7\text{ kg}\end{array}$

The answer is rounded off by dropping the last number, which is equal to $5$ , and the significant figure present on the left is even. Therefore, the last significant figure will remain unchanged. As a result, the final answer, after rounding off, is $76.6\text{ kg}$ .

Substitute the height as $1.88\text{ m}$ and the weight as $76.6\text{ kg}$ in equation $\left(1\right)$ .

$\begin{array}{c}\text{BMI}=\frac{76.7\text{ kg }}{{\left[1.88\text{ m}\right]}^2}\\ =\frac{76.7\text{ kg}}{3.534{\text{ m}}^2}\\ =21.7\text{ kg}/{\text{m}}^2\end{array}$

Therefore, the body mass index of the man is $21.7\text{ kg}/{\text{m}}^2$ .

Interpretation Introduction

Interpretation:

Whether the person is underweight, has a healthy weight, or is overweight:

Explanation of Solution

The standard rule of BMI states that if the value of BMI is between $18.5$ to $24.9$ , the person is considered to be healthy. The BMI value of this man is $21.6\text{ kg}/{\text{m}}^2$ ,which falls in this range. Therefore, the man is considered to be healthy.