y varies directly as the cube of x and inversely as the square of z.
If y is 30 when x is 2 and z is 4:
b) Find the equation that models this data.
c) Find the value of y when x is 3 and z is 2.
y = kx^3/z^2 -->y varies directly as the cube of x and inversely as the square of z.
Hopefully you can take it from here to answer your two-part question.
To find the equation that models this data, we need to combine the direct variation and inverse variation relationships.
The direct variation relationship can be written as:
y = k * x^3, where k is the constant of variation.
The inverse variation relationship can be written as:
y = k / z^2
To find the value of k, we can substitute the given values of x, y, and z into the equations:
For x = 2 and y = 30:
30 = k * 2^3
30 = k * 8
k = 30 / 8
k = 3.75
For x = 2 and z = 4:
30 = 3.75 / 4^2
30 = 3.75 / 16
30 = 0.234375
Therefore, the equation that models this data is:
y = 3.75 * x^3 / z^2
To find the value of y when x is 3 and z is 2, we can substitute these values into the equation:
y = 3.75 * 3^3 / 2^2
y = 3.75 * 27 / 4
y = 101.25 / 4
y = 25.3125
So, when x is 3 and z is 2, the value of y is approximately 25.3125.
To find the equation that models this data, we need to consider the direct variation of y with the cube of x and the inverse variation of y with the square of z.
Let's start with the direct variation relationship:
y varies directly as the cube of x can be expressed as y = kx^3, where k is the constant of variation.
Next, we consider the inverse variation relationship:
y varies inversely as the square of z can be expressed as y = k/z^2.
Now, we can combine the two relationships by multiplying the equations together:
(y = kx^3) * (y = k/z^2)
This gives us y^2 = kx^3 * k/z^2.
Simplifying this equation, we get:
y^2 = k^2 * (x^3/z^2)
Now, we can find the equation that models the data.
Given that y is 30 when x is 2 and z is 4:
Using these values in the equation, we have:
30^2 = k^2 * (2^3/4^2)
900 = k^2 * (8/16)
900 = k^2 * 1/2
900 = k^2/2
To solve for k, we can multiply both sides by 2 to eliminate the fraction:
1800 = k^2
Taking the square root of both sides, we find:
k = ±√1800
Now, substitute the value of k into one of the original equations to get the final equation that models the data. Let's use y = kx^3:
y = √1800 * x^3 (since k cannot be negative if y is positive)
Therefore, the equation that models the data is y = √1800 * x^3.
Now, let's find the value of y when x is 3 and z is 2.
Using the equation y = √1800 * x^3, substitute x = 3 and z = 2:
y = √1800 * 3^3
y = √1800 * 27
y ≈ 30.56
Therefore, when x is 3 and z is 2, y is approximately 30.56.