Suppose that y varies directly with x and inversely with zy=25 when x=35, and z=7. Write the equation that models the relationship. Then find y when x=12 and z=4
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Is there supposed to be a period after the first "z" ? (Before y = 25 ?)
If not, your question does not make sense to me.
To find the equation that models the relationship between y, x, and z, we need to determine the constant of proportionality. Let's start by setting up the equation using the given information:
y = k * (x / z)
We are told that y varies directly with x and inversely with z. Since y is directly proportional to x, we include x in the numerator. On the other hand, since y is inversely proportional to z, we include z in the denominator.
Now we can find the constant of proportionality (k) by substituting the given values of y, x, and z:
25 = k * (35 / 7)
Simplifying this equation, we have:
25 = 5k
Solving for k, we divide both sides of the equation by 5:
k = 5
Therefore, the equation that models the relationship is:
y = 5 * (x / z)
To find y when x = 12 and z = 4, we substitute these values into the equation:
y = 5 * (12 / 4)
y = 5 * 3
y = 15
So when x = 12 and z = 4, y is equal to 15.
To write an equation that models the relationship of y with x and z, we need to understand what it means for y to vary directly with x and inversely with z.
When two variables vary directly, it means that they have a constant ratio. In this case, we can say that y is directly proportional to x, represented as y ∝ x. Mathematically, we can express this relationship using a constant of variation, k, as y = kx.
When two variables vary inversely, it means that their product remains constant. In this case, we can say that y is inversely proportional to z, represented as y ∝ 1/z. Mathematically, we can express this relationship using a constant of variation, k', as y = k'/z.
Based on the given information, we know that when x = 35 and z = 7, y = 25. Now we can find the values of k and k' to form the equation.
Using the given values, we can substitute them into our equation y = kx and solve for k:
25 = k * 35
k = 25 / 35
k = 5/7
Similarly, using the given values, we can substitute them into our equation y = k'/z and solve for k':
25 = k' / 7
k' = 25 * 7
k' = 175
Therefore, the equation that models the relationship is:
y = (5/7) * x * (1/z)
Now, let's find the value of y when x = 12 and z = 4. We substitute these values into our equation:
y = (5/7) * 12 * (1/4)
y = (5/7) * 3
y = 15/7
So, when x = 12 and z = 4, y = 15/7.