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Height is a function of age. Suppose at birth the average male is 16 inches tall and grows 3 inches per year until age $12 .$ From age 12 to age 19 growth is 2 inches per year, assume there is no growth after age $19 .$ (a) Determine an equation which give the height $h$ as a function of age $a .$ What is the height at age (b) 9? (c) 15?

$h(a)=\left\{\begin{array}{cc}16+3 a & a<12 \\ 28+2 a & 12 \leq a \leq 19 \\ 66 & a>19\end{array}\right.$(b) 43 in(c) 58 in

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

03:18

00:42

If height (H) is a functio…

03:14

Childhood growth For child…

00:44

$h=3 a+28.6$A pediatri…

13:05

Using data collected at Ka…

this problem gives us information about the height of males as they grow from birth to adulthood. Our goal is to make a function h to show the height as a function of age. We're gonna call age A So is gonna be our variable. Looking over the problem, you can see that there are three different categories that we know about the height. So I'm going to make this a piece wise function with three parts. Now the first piece is up until they're 12. We know that there that says that males air 16 inches tall when they're born, and they grow it three inches per year. So they started 16. For every year of the age they gained three inches. But this is Onley true, up to 12 years old. So if somebody's two or three or eight, you can use that particular piece. However, things change a bit once they get over 12 from 12 Thio 19, they slow down now they're only gaining two inches a year. So in order to do this, we want to see how tall they are at age 12. So let's look at that first piece 16 plus three and at age 12. That's going to give me 52 inches. So for the second piece, we're starting at 52 inches. That's my starting points. I start this at age 12, and to that, I'm going to add two inches for every year they are above 12. So that's gonna give me two times a minus 12. So it's 13 13. That's gonna give me an extra two inches if they're 14, that gives me an extra four inches and so on. And the last pieces. If they're bigger than 19 at this point, we're assuming that they stopped growing. So it's a flat, constant height. So how tall are they at 19? Well, if I look at this second piece that starts at 52 inches 19 minus 12, it's sevens were adding an extra 14 inches to this, which gives me 66 inches. So once they're 19, they're at 66 inches and they're just going to stay there Now. I will say there's another way we can right this second piece. Uh, you don't need both. I'm just gonna kind of put this into, like, a little aside. Uh, what I've written here is perfectly valid. Some people don't like the parentheses. So if we distribute that to and combine our like terms, we can actually combine this into 28 plus to a So again, just another way to write it. We still have the same same restrictions on A. The numbers are gonna work out the same. It's just some people like that a little bit cleaner, some people before it this way, because you can really see where those numbers are coming from. But regardless of which one you use, they get used the same way. So we're gonna find to heights. First, we're gonna find the height of a nine year old and then the height of a 15 year old. Okay, nine year old first nine is less than 12. So we're going to use that blue equation that first case 16 plus three times nine. Doing that out. That gives us ah, height of 43 inches. Okay, Now, what about H of 15? Well, 15 falls into our second category and again, doesn't matter which one to use. I'm just going to use the one that I marked in red. That's the one I did first. So I have 52 plus two times three. So I get three extra years of that two inches a year growth, so that gives us a total of 58 inches.

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