The quantity y varies directly with the square of x and inversely with z. When x is 9 and z is 27, y is 6. What is the constant of variation?

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The quantity y varies directly with the square of x and inversely with z. When x is 9 and z is 27, y is 6. What is the constant of variation?

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Answer 1 · 2017-05-22T05:07:50

The constant of variation is $k = 2$ $k=2$ .

Explanation:

To say that a variable "varies directly" with some quantity, we mean that the variable scales with that quantity. In this example, that means the scaling of $y$ $y$ is "in sync" with the scaling of $x^{2}$ $x^2$ (i.e. when $x^{2}$ $x^2$ doubles, $y$ $y$ also doubles).

We're also given that $y$ $y$ varies inversely with $z$ $z$ , meaning that when $z$ $z$ doubles, $y$ $y$ gets halved.

We can take the information given and form it into a single equation like this:

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

The $k$ $k$ is the constant of variation we seek. Plugging in the given values of $x$ $x$ , $y$ $y$ , and $z$ $z$ into this equation, we get

$6 = k \cdot \frac{9^{2}}{27}$ $6=k*(9^2)/(27)$

$6 = k \cdot \frac{81}{27}$ $6=k*81/27$

$6 = k \cdot 3$ $6=k*3$

$2 = k$ $2=k$

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The quantity y varies directly with the square of x and inversely with z. When x is 9 and z is 27, y is 6. What is the constant of variation?

1 Answer

Explanation:

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