Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Feb 29, 2024

Find $z$ when $x=5$ if $z$ varies directly as $x^{2}$ and $z=8$ when $x=2$ .

STEP 1

Assumptions
1. The variable $z$ varies directly as $x^{2}$ .
2. When $x=2$ , $z=8$ .
3. We need to find the value of $z$ when $x=5$ .

STEP 2

Since $z$ varies directly as $x^{2}$ , we can write the relationship as:
$z = kx^{2}$
where $k$ is the constant of proportionality.

STEP 3

We need to find the value of $k$ using the information that $z=8$ when $x=2$ .
$8 = k \cdot 2^{2}$

STEP 4

Solve for $k$ by dividing both sides of the equation by $2^{2}$ .
$k = \frac{8}{2^{2}}$

STEP 5

Calculate the value of $k$ .
$k = \frac{8}{4} = 2$

STEP 6

Now that we have the value of $k$ , we can use it to find $z$ when $x=5$ .
$z = k \cdot 5^{2}$

STEP 7

Substitute the value of $k$ into the equation.
$z = 2 \cdot 5^{2}$

STEP 8

Calculate the value of $z$ .
$z = 2 \cdot 25$

STEP 9

Finish the calculation to find $z$ .
$z = 2 \cdot 25 = 50$
Therefore, when $x=5$ , $z=50$ .

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