Determine the equation of combined variation. Then solve for the missing value.
x varies directly with y and inversely with z.
x = 5 when y = 10 and z = 5.
Find x when y = 20 and z = 10.
x = ky/z
5 = k(10)/(5) ==> k = 5/2
x(20,10) = (5/2)(20)/(10) = 5
To determine the equation of combined variation, we need to combine the direct variation and inverse variation equations.
Let's start by stating the equation of direct variation:
x = ky
Where k is the constant of variation.
Next, let's state the equation of inverse variation:
x = k/z
Now, since x varies directly with y and inversely with z, we can express this as:
x ∝ yz
To remove the proportionality symbol (∝), we introduce a constant of variation. Let's call it k2:
x = k2 * yz
Now, we can solve for k2. Given that when y = 10 and z = 5, x = 5, we can substitute these values into the equation:
5 = k2 * (10 * 5)
5 = k2 * 50
Dividing both sides by 50:
k2 = 5/50
k2 = 1/10
Now, we can use this value of k2 to find x when y = 20 and z = 10:
x = (1/10) * (20 * 10)
x = (1/10) * 200
x = 20
Therefore, when y = 20 and z = 10, x = 20.