Now, let's deal with joint variation, where one is never enough. In fact, one seriously depends on two others, making a mathematical trio. Turns out it's not so fancy-shmancy: joint variation is like direct variation, but it involves more than one variable. The general form looks like this:
y = kxz
...where y varies directly with x, and y also varies directly with z. In other words, y varies directly with the product of xz. The constant of variation is still our good buddy, k.
Typically, we use this formula to find the area of a rectangle:
A = lw
Here k = 1.
If either the length or the width increases, the area increases. If either the length or the width decreases, the area decreases. Just think of lots of moving parts, geometrically speaking.
Sample Problem
If y varies jointly with the product of x and z, and y = 72 when x = 4 and z = 9, find y when x = 5 and z = 6.
First, we bust out our general form for joint variation.
y = kxz
Plug in our values to find k.
72 = k(4)(9)
Solve for k.
72 = 36k
k = 2
Plug k back into our general form.
y = 2xz
Finally, find y when x = 5 and z = 6.
y = 2(5)(6)
y = 60
Sample Problem
If y varies jointly with the product of x and z, and y = 40 when x = 1 and z = 2, find y when x = 3 and z = 4.
You know the drill. Start with the general formula.
y = kxz
Plug in everything we know.
40 = k(1)(2)
Now we can find k.
40 = 2k
k = 20
Throw k back into our general equation.
y = 20xz
Last but definitely not least, find y when x = 3 and z = 4.
y = 20(3)(4)
y = 240
Sample Problem
If y varies jointly with the product of x and z, and y = 30 when x = 10 and z = 6, find y when x = 7 and z = 2.
Not to sound like a broken record here, but once again, we whip out the general formula.
y = kxz
Throw in our values for x, y, and z, then stir the pot and heat at 350°.
30 = k(10)(6)
Solve for k.
30 = 60k
k = 0.5
Plug k back into our general form.
y = 0.5xz
Demolish this problem by finding y when x = 7 and z = 2.
y = 0.5(7)(2)
y = 7