Joint Variation

Question 1

Given $y \propto xz$ and $y = 24$ when $x = 2$, $z = 3$, find $k$.

Accepted Answer

4 — $k = \frac{y}{xz} = \frac{24}{2 	imes 3} = \frac{24}{6} = 4$.

Question 2

Given $y = 4xz$, find $y$ when $x = 5$, $z = 2$.

Accepted Answer

40 — $y = 4(5)(2) = 40$.

Question 3

Given $y \propto \frac{x}{z}$ and $y = 6$ when $x = 4$, $z = 2$, find $k$.

Accepted Answer

3 — $k = \frac{yz}{x} = \frac{6 	imes 2}{4} = \frac{12}{4} = 3$.

Question 4

Given $y = \frac{3x}{z}$, find $y$ when $x = 8$, $z = 4$.

Accepted Answer

6 — $y = \frac{3(8)}{4} = \frac{24}{4} = 6$.

Question 5

If $y$ varies directly as $x$ and inversely as $z$, then

Accepted Answer

$y = \frac{kx}{z}$ — Direct with $x$ means $x$ on top; inverse with $z$ means $z$ on the bottom.

Question 6

Given $y \propto xz^2$ and $y = 36$ when $x = 3$, $z = 2$, find $k$.

Accepted Answer

3 — $k = \frac{y}{xz^2} = \frac{36}{3 	imes 4} = \frac{36}{12} = 3$.

Question 7

Given $y = kxz$ with $y = 60$ when $x = 3$, $z = 4$, find $y$ when $x = 2$, $z = 5$.

Accepted Answer

50 — $k = \frac{60}{12} = 5$, so $y = 5(2)(5) = 50$.

Question 8

“$y$ is directly proportional to $x$ and $z$ jointly” is written as

Accepted Answer

$y = kxz$ — Joint direct variation with $x$ and $z$ gives $y = kxz$.

Question 9

Given $y \propto \frac{x}{z^2}$ and $y = 2$ when $x = 8$, $z = 2$, find $y$ when $x = 18$, $z = 3$.

Accepted Answer

2 — $k = \frac{yz^2}{x} = \frac{2 	imes 4}{8} = 1$, so $y = \frac{18}{3^2} = \frac{18}{9} = 2$.

Question 10

In $y = kxz$, if $x$ is doubled and $z$ is tripled, then $y$ becomes

Accepted Answer

6 times — $y$ changes by a factor of $2 	imes 3 = 6$.