Mathematics: The variable $x$ varies directly as the square of $y$, and $y$ varies directly as the cube of $z$. If $x$ equals $-16$ when $z$ equals 2, what is the value of $x$ when $z$ equals $rac{1}{2}$?

The variable $x$ varies directly as the square of

21hendlill

19.01.2022

If varies directly as y^2 , then there is some constant for which

x=ay^2

Similarly, there is some constant such that

y=bz^3

Given that x=-16 when z=2 , we have

$\begin{cases}-16=ay^2\\y=8b\end{cases}\implies-16=a(8b)^2\implies ab^2=-\dfrac14$

Now when $z=\frac12$ , we get

$\begin{cases}x=ay^2\\y=\frac b8\end{cases}\implies x=a\left(\dfrac b8\right)^2=\dfrac{ab^2}{64}=\dfrac{-\frac14}{64}=\boxed{-\dfrac1{256}}$

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Kpthegoat

28.03.2023

Angle DAB is 29°.

Step-by-step explanation:

The lines in the two angles show that they are equal.

So angle DAB is equal to angle CAD.

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