QuestionSuppose $z$ varies directly with $x$ and inversely with the square of $y$ . If $z=8$ when $x=4$ and $y=3$ , what is $z$ when $x=8$ and $y=6$ ? $z=$ $\square$ Next Question

Studdy Solution

STEP 1

1. The variable $z$ varies directly with $x$ .
2. The variable $z$ varies inversely with the square of $y$ .
3. When $z = 8$ , $x = 4$ , and $y = 3$ .
4. We need to find $z$ when $x = 8$ and $y = 6$ .

STEP 2

1. Establish the relationship between $z$ , $x$ , and $y$ .
2. Determine the constant of variation.
3. Use the constant to find the new value of $z$ .

STEP 3

Establish the relationship between $z$ , $x$ , and $y$ .
Since $z$ varies directly with $x$ and inversely with the square of $y$ , we can express this relationship as:
$z = k \cdot \frac{x}{y^2}$
where $k$ is the constant of variation.

STEP 4

Determine the constant of variation $k$ .
Substitute the known values $z = 8$ , $x = 4$ , and $y = 3$ into the equation:
$8 = k \cdot \frac{4}{3^2}$
Simplify the equation:
$8 = k \cdot \frac{4}{9}$
Solve for $k$ by multiplying both sides by $\frac{9}{4}$ :
$k = 8 \cdot \frac{9}{4}$ $k = 18$

STEP 5

Use the constant $k$ to find the new value of $z$ when $x = 8$ and $y = 6$ .
Substitute $k = 18$ , $x = 8$ , and $y = 6$ into the equation:
$z = 18 \cdot \frac{8}{6^2}$
Simplify the equation:
$z = 18 \cdot \frac{8}{36}$ $z = 18 \cdot \frac{2}{9}$ $z = 4$
The value of $z$ is:
$\boxed{4}$

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