For the following exercises, write an equation describing...

For the following exercises, write an equation describing the relationship of the given variables.
$y$ varies jointly as $x$ and $z$ and inversely as the square root of $w$ and the square of $t .$ When $x=3, z=1, w=25,$ and $t=2,$ then $y=6$

Key Concepts

Constant of Proportionality

The constant of proportionality is a fixed number that links the direct and inverse relationships in a variation equation. It is determined by substituting known values into the relationship and is essential for establishing the precise mathematical formula connecting the variables.

Joint Variation

Joint variation occurs when a quantity varies directly as the product of two or more other quantities. In this type of relationship, changing any of the independent variables will proportionally affect the dependent variable since they are multiplied together.

Inverse Variation

Inverse variation describes a relationship in which a quantity varies inversely as one or more other quantities. This means that as the independent variable increases, the dependent variable decreases in such a way that the product (or quotient) remains constant, highlighting a reciprocal connection between the variables.

Transcript

00:02 All right.

00:03 So we have a super mondo problem here where we have both joint variation and inverse variation.

00:10 And it's not just simple inverse of a value.

00:13 It's a square root of w and a square of t.

00:15 So it's really exciting stuff.

00:18 I've got why, excuse me, y varying jointly as x and z.

00:22 So i'm going to set that up like i would a normal varying jointly problem.

00:26 Y equal since it's varying.

00:28 And then jointly says have a k because i need a constant.

00:32 And then multiply it by what it's varying jointly with, so x and z.

00:38 But at the same time, if you just ignore that part of the sentence, it's saying y varies inversely as the square root of w and the square of t.

00:46 So inversely means go ahead and divide your k by these values, and i've got the square root of w and the square of t.

00:57 So that's t squared.

00:59 Okay, this is great if i want a generic formula, but that's not telling me anything about the specific of this problem, which are given to me in the x, y, z, and all those values they give me.

01:09 So i've got x equals three, z equals one, w equals 25, t equals two, and y equals six.

01:23 So all of these guys, all these values need to go into this generic equation so that i can find k.

01:32 And once i have k, i can then rewrite the equation to specifically represent the relationship between k, the constant x, z, square root of w, t squared, and y.

01:45 So i'm going to go ahead and do that now.

01:47 I'm going to insert all these values.

01:49 So y is six...

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