Definition: A quantity varies jointly as two (or more) other quantities if it is directly proportional to their product.
[ y = kxz ]
Key phrase to look for on a Kuta worksheet: "varies jointly as" or "jointly proportional to".
Example: The area of a triangle (A) varies jointly as its base (b) and height (h).
[ A = k \cdot b \cdot h ] (In geometry, we know (k = \frac12), but in algebra problems, you solve for (k) first).
If $y$ varies inversely as $x$, then as $x$ goes up, $y$ goes down. joint and combined variation worksheet kuta
Combined variation involves a combination of direct and inverse variation. It can be represented by the equation:
$$y = \frackxz$$
or
$$y = kx + \fracmz$$
Let’s walk through a problem you would find on a joint and combined variation worksheet kuta.
Problem Type 1 (Joint): "If y varies jointly as x and z, and y = 24 when x = 4 and z = 2, find y when x = 10 and z = 5."
Solution:
Problem Type 2 (Combined): "If y varies directly as x and inversely as z, and y = 10 when x = 5 and z = 2, find y when x = 20 and z = 4." Definition: A quantity varies jointly as two (or
Solution:
Problem:
The volume ( V ) of a gas varies directly as the temperature ( T ) (in Kelvin) and inversely as the pressure ( P ). If ( V = 200 ) cm³ when ( T = 300 ) K and ( P = 2 ) atm, find ( V ) when ( T = 360 ) K and ( P = 3 ) atm.
Solution:
Step 1: ( V = \frack \cdot TP )
Step 2: ( 200 = \frack \cdot 3002 ) → ( 200 = 150k ) → ( k = \frac200150 = \frac43 )
Step 3: ( V = \frac(4/3) \cdot TP )
Step 4: ( V = \frac(4/3) \cdot 3603 = \frac4803 = 160 )
Answer: ( V = 160 ) cm³
The equation for joint variation is $y = kxz$, where $y$ is the variable, $x$ and $z$ are the variables it varies with, and $k$ is the constant of variation. Combined variation involves a combination of direct and
The equation for combined variation is $C = k \fracnw^2$, where $C$ is the cost, $n$ is the number of TVs, $w$ is the number of workers, and $k$ is the constant of variation.
Now that you know (k=4), rewrite the equation: (y = 4xz).