The body mass index (BMI) of a person is defined by B(m, h)=(m)/(h^2) where m is the person's ma

The body mass index (BMI) of a person is defined by B(m,...

The body mass index (BMI) of a person is defined by $$ B(m, h)=\frac{m}{h^{2}} $$ where $m$ is the person's mass (in kilograms) and $h$ is the height (in meters). Draw the level curves $B(m, h)=18.5$ $B(m, h)=25, B(m, h)=30,$ and $B(m, h)=40 .$ A rough guideline is that a person is underweight if the BMI is less than $18.5 ;$ optimal if the BMI lies between 18.5 and $25 ;$ overweight if the BMI lies between 25 and $30 ;$ and obese if the BMI exceeds $30 .$ Shade the region corresponding to optimal BMI. Does someone who weighs 62 kg and is 152 $\mathrm{cm}$ tall fall into this category?

The body mass index (BMI) of a person is defined by
$$
B(m, h)=\frac{m}{h^{2}}
$$
where $m$ is the person's mass (in kilograms) and $h$ is the
height (in meters). Draw the level curves $B(m, h)=18.5$
$B(m, h)=25, B(m, h)=30,$ and $B(m, h)=40 .$ A rough
guideline is that a person is underweight if the BMI is less than
$18.5 ;$ optimal if the BMI lies between 18.5 and $25 ;$ overweight
if the BMI lies between 25 and $30 ;$ and obese if the BMI
exceeds $30 .$ Shade the region corresponding to optimal BMI.
Does someone who weighs 62 kg and is 152 $\mathrm{cm}$ tall fall into
this category?

Key Concepts

Level Curves

A level curve of a function of two variables is the set of points for which the function has a constant value. They provide insight into the behavior of the function by illustrating its structure and how the output value changes with varying inputs.

Implicit Equations

When level curves are determined, they often lead to equations that define the relationship between the variables implicitly. Solving these equations can reveal the geometric nature of the curves, such as parabolic or elliptical shapes, which are then plotted for analysis.

Inequalities and Region Shading

Inequalities are used to define regions where the function's value falls within a particular range. Shading these regions on a graph helps visualize areas that meet specified criteria, allowing for clear categorization of different outcomes or conditions.

Body Mass Index (BMI)

BMI is a specific example of a function that relates mass and height through the formula m divided by the square of h. It is used as an indicator to classify individuals into categories like underweight, optimal, overweight, and obese, serving as an applied example of how level curves and regions associated with function values can be interpreted.

Transcript

00:01 So they give us this formula to calculate bmi where mass is in kilograms and our height is in meters.

00:08 So the first thing they tell us to do is draw the level curves for these bmis.

00:13 So all we're going to need to do is set these equations equal to.

00:20 Well, so first we can just say this is equal to some value c.

00:23 So i'll do this over here.

00:24 So it would be like c is equal to m over h squared.

00:28 And if we think about it, we can't have a mass less.

00:31 Than zero and we can't have a height less than zero.

00:33 So we can just go ahead and multiply over by h squared and we won't have to worry about this being like equal to zero or anything.

00:39 So we would just have c h squared is equal to m.

00:43 So it looks like this is going to just be some quadratic equation and remember since both m and h are going to be positive this is just going to be in the first quadrant.

00:52 So all we really need to do is draw the first quadrant here, where this over here is going to be for our input and then over here is m for our output.

01:05 All right.

01:08 So now we can go ahead and just plug in.

01:13 Well, these are all of our cs here.

01:20 So for this first one, so i'll do this in red.

01:26 So if h was zero, so this equation will just give us.

01:31 So maybe i should write this like below.

01:33 It would be 18 .5.

01:35 H squared is equal to m so we would just be starting from zero here and then if we were to plug in like one meter this is going to give us just a height of 18 .5s would be around here and then if we were to plug in like two meters so that would be 18 .5 times four so that would be 74.

02:02 So it would be up here and so then we can go ahead and draw this level curve like that and it just kind of keeps on going up.

02:13 Alright, now for the next, actually maybe i should make this a little bit smaller and you'll see why because we'll need to really be squeezing in a lot.

02:23 So the one meter and the two meters for height, we can keep those, but over here i want to probably squish this down a little bit more.

02:33 So i'll say like 18 .5 is here and then one, two, so 74 is probably around here or so.

02:47 So we have this, and then we have that there.

02:50 So this is going to be our level curve.

02:54 Or c is equal to, or maybe i should say instead, bmi is equal to 18 .5.

03:03 Now for the next one i'll do here in blue.

03:06 So this will give us the equation 25h squared is equal to m.

03:10 So it's going to start from the same spot.

03:12 But now when we plug in 1 for h, it's going to get us at 25.

03:17 So we're going to start here.

03:20 So it would be here.

03:21 And then if we plug in 2, well, that's going to give us 4.