To find the missing values in the table, we need to use the given relationship between y, x, and w.
The relationship states that y varies directly as the cube of x and inversely as the square root of w. Therefore, we can write the equation as:
y = k * (x^3 / sqrt(w))
To find the constant of variation, k, we can use one of the given data points. Let's use the third row: w = 4.66, x = 2.75, and y = 3.87.
3.87 = k * (2.75^3 / sqrt(4.66))
Now, we can solve for k:
k = 3.87 * sqrt(4.66) / (2.75^3)
Using this value of k, we can find the missing values.
1. For the first row: w = 1.27 and y = 3.05, we can find x:
3.05 = k * (x^3 / sqrt(1.27))
x^3 = 3.05 * sqrt(1.27) / k
x = (3.05 * sqrt(1.27) / k)^(1/3)
2. For the second row: x = 5.66 and y = 1.93, we can find w:
1.93 = k * (5.66^3 / sqrt(w))
sqrt(w) = (5.66^3 / (k * 1.93))
w = (5.66^6 / (k * 1.93^2))
3. For the fourth row: w = 7.07 and x = 1.56, we can find y:
y = k * (1.56^3 / sqrt(7.07))
By substituting the value of k calculated earlier, we can find all the missing values.