The problem states that $y$ varies directly as $x$ and inversely as the square of $z$. We are given that when $x=8$, $y=4$ and $z=2$. We are asked to find: a. The value of the constant of proportionality $k$. b. The value of $y$ when $x=18$ and $z=3$. c. The value(s) of $z$ when $x=25$ and $y=2$.

1. Problem Description

The problem states that $y$ varies directly as $x$ and inversely as the square of $z$. We are given that when $x=8$, $y=4$ and $z=2$. We are asked to find:

a. The value of the constant of proportionality $k$.

b. The value of $y$ when $x=18$ and $z=3$.

c. The value(s) of $z$ when $x=25$ and $y=2$.

2. Solution Steps

a. Since $y$ varies directly as $x$ and inversely as the square of $z$, we can write the relationship as:

$y = k \frac{x}{z^2}$

We are given that $x=8$, $y=4$, and $z=2$. Substituting these values into the equation, we get:

$4 = k \frac{8}{2^2}$

$4 = k \frac{8}{4}$

$4 = 2k$

$k = \frac{4}{2} = 2$

Therefore, $k=2$.

b. Now we want to find the value of $y$ when $x=18$ and $z=3$. We have $k=2$, so the equation is:

$y = 2 \frac{x}{z^2}$

Substituting $x=18$ and $z=3$, we get:

$y = 2 \frac{18}{3^2}$

$y = 2 \frac{18}{9}$

$y = 2(2) = 4$

Therefore, $y=4$.

c. Finally, we want to find the value(s) of $z$ when $x=25$ and $y=2$. Again, we use the equation:

$y = 2 \frac{x}{z^2}$

Substituting $x=25$ and $y=2$, we get:

$2 = 2 \frac{25}{z^2}$

Divide both sides by 2:

$1 = \frac{25}{z^2}$

$z^2 = 25$

Taking the square root of both sides, we get:

$z = \pm \sqrt{25}$

$z = \pm 5$

Therefore, $z=5$ or $z=-5$.

3. Final Answer

a. $k=2$

b. $y=4$

c. $z=5$ or $z=-5$