Question

6. The graph on the next page is a doctor's chart for calculating body mass index (BMI) percentiles for girls aged 2 to 20 years (I got this chart from www.cdc.gov/growthcharts/ clinical_charts.htm). The r-axis is age (in years), and the y-axis is BMI (in kg/m2). BMI is calculated as weight divided by height squared. The different curves represent different percentiles, denoted by the small number on the right of each curve. If a girl is in the 95 percentile (top curve), this means that her BMI is higher than 95% of other girls her age. th (a) Imagine you are a doctor and a 15 year old girl has come in for her yearly check up You measure her height as 1.55 m and her weight as 57.67 kg. What is her BMI? What percentile does this put her in? (b) Consider the 5th percentile curve (bottom curve) i. Describe (using words like "increases" and "decreases") how BMI changes with age. ii. At what age do critical points and inflection points occur? (Make sure you specify iii. Most doctors don't talk to their patients about inflection points. As a doctor, how Make sure you consider the concavity of the curve. which are critical points and which are inflection point:s will you describe to your patient what will likely happen to her BMI between ages .) 10 and 20 years? (c) Consider the 85th percentile curve and identify the ages at which critical points and inflection points occur. How do these compare to what you found for the 5th percentile curve in part (b)?

Answer 1

a. As described BMI of the 15 year old girl may be calculated using the formula weight in kg divided by square of height in m. So for the girl it will be 57.67/(1.55)²= 24. This puts her in the 85 percentile because in the 85 percentile curve the BMI attains a value of 24 at the age of 15.

b. i. If we analyze the 5th percentile, we find at first BMI decreases with age and then it starts increasing. Initially till approx 3 years the rate of decrease of BMI with age is rapid, after which the BMI still decreases with age but with lower rate. At approx 6-7 years of age a plateau is reached at which there is negligible change of BMI with age. After the age of 7, BMI starts increasing, initially at a lower rate till at the age of 11. Till this point the graph is having a concave upwards. After the age of 11, the BMI increases at a faster rate and the graph becomes concave downwards.

ii. The critical point is that point in the graph, where the BMI is lowest and the slope becomes 0 or a plateau is reached. Such a point is achieved at the age of 7, when the BMI is the lowest. Therefore the age of 7 is the critical point. The inflection point is the point at which the graph transitions from upward concavity to the downward concavity. It is the point when the slope becomes increasing from decreasing. Or in other words when the rate of increase starts becoming rapid. We see that after the age of 11 this is attained. Therefore age 11 is the inflection point.

iii. As a doctor, I would try to explain to my patient that the age of 10-20 will see her BMI increase at a very rapid rate.

C. At the 85th percentile, the critical age is at 4.5 years of age and the inflection point is 6.5 years of age. Therefore in comparison with the 5th percentile girls, this group will start the increase in BMI at a very early age so that they will have more BMI than the BMI of 5th percentile at a given age.