Variasi gabungan

(Video) Joint Variation

by ThinkwellVids

Here's a video by ThinkwellVids explaining joint variation.

He gives the following problem: "The area $a$ of a triangle varies jointly as the base $b$ and the height $h$".

Varies jointly means that the function is varied by both a constant and the product of two variables. Written out as a function, this looks like: $$y=kxz$$ Where $k$ is the constant, and $x$ and $x$ are variables.

He then adds some more information to the problem, "when the $a=20$, $b=5$, and $h=8$, find $b$ when $a=60$ and $h=6$".

First, we're find the constant. $$\begin{align} a&=kbh\\ 20&=k(5)(8)\\ 20&=40k\\ k&=\frac12 \end{align}$$

You may recall that the area of a triangle is $a=\frac{1}{2}bh$, so we have calculated our constant while using something we already know as an example!

Now, we're going to find $b$ when $a=60$ and $h=6$. $$\begin{align} a&=kbh\\ 60&=(\frac12) b (6)\\ 60&= 3 b\\ b&=2 \end{align}$$

**Joint Variation ** $\\$ Joint variation is a type of variable where one variable depends on two or more variables. $\\$ If $z$ varies jointly in proportion to $x$ and $y$, the jointly proportional equation will be in the form $z=kxy$. $\\$ In other words, $z$ is directly proportional to both variables $x$ and $y$.

Let’s take a look at an example. $\\$

Since we know that $z$ is jointly proportional to $x$ and $y$, we can write the equation in the form $z=kxy$. $\\$ We are given that when $z=24$, $x=2$ and $y=3$. $$\begin{align} z&=kxy \\ 24 &= k(2)(3) \\ 24 &= 6k \\ k &= \dfrac{24}{6} \\ k &= 4 \\ \end{align}$$ Now that we found $k$, we can find $z$ when $x=4$ and $y=5$. $$\begin{align} z&=kxy \\ z &= 4xy \\ z &= 4(4)(5) \\ z &= 16(5) \\ z &= 80 \\ \end{align}$$