# 5.8 Modeling using variation  (Page 4/14)

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$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the square of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$y=5{x}^{2}$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the square root of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the cube of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when $\text{\hspace{0.17em}}x=36,\text{\hspace{0.17em}}y=24.$

$y=10{x}^{3}$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the cube root of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the fourth power of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$y=6{x}^{4}$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely as the square of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

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

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely as the cube of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely as the fourth power of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$y=\frac{81}{{x}^{4}}$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely as the square root of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely as the cube root of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$y=\frac{20}{\sqrt[3]{x}}$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly with $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ and when $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as and $\text{\hspace{0.17em}}w\text{\hspace{0.17em}}$ and when then $\text{\hspace{0.17em}}y=100.$

$y=10xzw$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as the square of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and the square of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ and when $\text{\hspace{0.17em}}x=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=4,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=72.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and the square root of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ and when $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=25,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=100.$

$y=10x\sqrt{z}$

$\text{\hspace{0.17em}}y$ varies jointly as the square of $\text{\hspace{0.17em}}x$ the cube of $\text{\hspace{0.17em}}z$ and the square root of $\text{\hspace{0.17em}}W.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=1,z=2,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}w=36,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=48.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as and inversely as $\text{\hspace{0.17em}}w.\text{\hspace{0.17em}}$ When and $\text{\hspace{0.17em}}w=6,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=10.$

$y=4\frac{xz}{w}$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as the square of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and the square root of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ and inversely as the cube of $\text{\hspace{0.17em}}w\text{.\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=3,z=4,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}w=3,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=6.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ and inversely as the square root of $\text{\hspace{0.17em}}w\text{\hspace{0.17em}}$ and the square of $\text{\hspace{0.17em}}t\text{\hspace{0.17em}.}$ When $\text{\hspace{0.17em}}x=3,z=1,w=25,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}t=2,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=6.$

$y=40\frac{xz}{\sqrt{w}{t}^{2}}$

## Numeric

For the following exercises, use the given information to find the unknown value.

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=12.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ wneh $\text{\hspace{0.17em}}x=20.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the square of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=16.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $x=8.$

$y=256$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the cube of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ then Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=4.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the square root of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=16,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=4.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=36.$

$y=6$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the cube root of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=125,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=15.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=1,000.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely with $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=2.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=1.$

$y=6$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely with the square of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=4,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=3.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=2.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely with the cube of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=1.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=1.$

$y=27$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely with the square root of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=64,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=12.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=36.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely with the cube root of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=27,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=5.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=125.$

$y=3$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as When $\text{\hspace{0.17em}}x=4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=2,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=16.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=3.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as When $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ $z=1,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}w=12,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=72.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=1,\text{\hspace{0.17em}}$ $z=2,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}w=3.$

$y=18$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and the square of $\text{\hspace{0.17em}}\mathrm{z.}\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=4,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=144.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=5.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as the square of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and the square root of $\text{\hspace{0.17em}}z.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=9,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=24.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=25.$

$y=90$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ and inversely as $\text{\hspace{0.17em}}w.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=5,\text{\hspace{0.17em}}$ and then $y=4.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=3\text{\hspace{0.17em}}$ and and

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as the square of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and the cube of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ and inversely as the square root of $\text{\hspace{0.17em}}w\text{.\hspace{0.17em}}$ When $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ $z=2,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}w=64,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=12.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=1,\text{\hspace{0.17em}}$ $z=3,\text{\hspace{0.17em}}$ and

$y=\frac{81}{2}$

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