Body Mass Index The BMI is usually used as a guideline to...

Body Mass Index The BMI is usually used as a guideline to determine whether a person is overweight or underweight. For example, according to the Centers for Disease Control, a 12 -year-old boy is at risk of being overweight if his BMI is between 21 and 24 and is considered overweight if his BMI is above 24 . A 13 -year-old boy is at risk of being overweight if his BMI is between 22 and 25 and is overweight if his BMI is above 25 . (a) At age 12 , a boy's weight was 50 kilograms and his height was $1.55$ meters. Find his BMI and determine whether he is overweight or at risk of being overweight. (b) Suppose that the boy's weight was increasing at the rate of 7 kilograms per year and his height was increasing at the rate of 5 centimeters per year. Find the rate of change of his BMI at age 12 or $b^{\prime}(12)$. (c) Use the values $b(12)$ and $b^{\prime}(12)$ to estimate $b(13)$, the BMI of the child at age 13 . Is the child at risk of being overweight when he turns $13 ?$

Body Mass Index The BMI is usually used as a guideline to determine whether a person is overweight or underweight. For example, according to the Centers for Disease Control, a 12 -year-old boy is at risk of being overweight if his BMI is between 21 and 24 and is considered overweight if his BMI is above 24 . A 13 -year-old boy is at risk of being overweight if his BMI is between 22 and 25 and is overweight if his BMI is above 25 .
(a) At age 12 , a boy's weight was 50 kilograms and his height was $1.55$ meters. Find his BMI and determine whether he is overweight or at risk of being overweight.
(b) Suppose that the boy's weight was increasing at the rate of 7 kilograms per year and his height was increasing at the rate of 5 centimeters per year. Find the rate of change of his BMI at age 12 or $b^{\prime}(12)$.
(c) Use the values $b(12)$ and $b^{\prime}(12)$ to estimate $b(13)$, the BMI of the child at age 13 . Is the child at risk of being overweight when he turns $13 ?$

Key Concepts

Body Mass Index (BMI)

BMI is a numerical value calculated from a person's weight and height and is widely used as an indicator of body fatness and general health. It is computed by dividing weight by the square of the height and provides a quick assessment to determine if a person is underweight, normal weight, or overweight. This metric is especially useful in public health to monitor population trends and assess health risks related to body weight.

Related Rates

Related rates problems involve determining the rate at which one quantity changes with respect to another, given the rate of change of a related variable. In the context of this problem, both weight and height are changing over time, and their changes affect BMI. Solving related rates problems often requires setting up an equation that links the related variables and then differentiating with respect to time to find the rate of change of the quantity of interest.

Differentiation (Quotient Rule and Chain Rule)

When dealing with functions that are ratios of two differentiable functions, the quotient rule is a key technique used to find the derivative. The chain rule is also frequently applied when a variable depends on another variable that is itself changing with time. In the BMI problem, both the quotient rule and chain rule are utilized to differentiate the BMI formula with respect to time, yielding the rate at which BMI changes as weight and height increment over time.

Linear Approximation

Linear approximation, also known as differential approximation, is a method used to estimate the value of a function at a point close to a known value. This technique uses the function’s derivative to extrapolate the function’s behavior. In this problem, once the current BMI value and its rate of change are known, linear approximation helps estimate the BMI at a later time, providing a practical application of calculus to predict future outcomes based on current trends.

Transcript

00:03 So today we're going to be looking at the publics and we are going to find the body, the bmi of a 12 -year -old relay, and then also the rate of change of the bmi.

00:15 So we're going to start out, we have our function.

00:18 Bmi is calculated by taking the weight in kilograms over the height squared meters.

00:24 Now, looking ahead, we are going to have to find the derivative of this function.

00:28 So i just want to point out that the denominator of this function is h of t squared in the way we take the derivative.

00:35 Of a function squared is using the extended power rule, which is really just an application of the chain rule.

00:43 So that's 2h of t times h prime of t.

00:46 So we will need that in part b.

00:48 But first for part a, we got told that the weight is 50 kilograms and the height is 1 .5 meters.

00:56 So when we do that square, we get 20 .8117.

01:01 And the question is whether or not the boy is at risk.

01:06 For being overrate or at risk of being overrate.

01:10 But since the bmi is less than 21, according to the guidelines, stated not at risk or overrate.

01:21 Now in the next problem, we are asked to find the rate of change of this same individual at 12 years old.

01:31 So we will need to find the derivative of our function using the quotient rule.

01:36 So the derivative of the quotient rule says we take this derivative of the numerator and we're going to be evaluating all of this at 12.

01:43 The derivative of the numerator times the denominator, that's all squared, minus the numerator times the derivative of the denominator, which is two times h of 12, h prime of 12 as we just saw all over, oops, all over the denominator squared.

02:08 So that's h of the 12 to the fourth.

02:15 And we can substitute in all the values that we are given...

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