Answer:
I) Relationship between P, Q, and Z: P = 5/9 * (Q^2 / Z^3)
II) Z is approximately 1.67 when P = 3 and Q = 5
Explanation:
For II), did you mean to write Z when P = 3 and Q = 5.
If you meant to write when P = -3 and Q = -5, write it in the comments and I'll edit my answer.
I)
- We're looking for a joint variation equation.
- First, we need to look at the general equation for direct variation and inverse variation, which will help us when looking for the join variation equation.
Direct variation equation:
The general equation for direct variation is given by:
y = kx, where
- k is the constant of proportionality.
Inverse variation equation:
The general equation for inverse variation is given by:
y = k / x, where
- k is the constant of proportionality.
Since P varies directly as the square of Q and inversely as the cube of Z, we can represent this algebraically with the following equation:
P = k * (Q^2 / Z^3)
Determining the constant of proportionality, k:
Before we can determine the relationship between P, Q, and Z, we need to determine the constant of proportionality, k by substituting 5 for P, 3 for Q, and 1 for Z:
5 = k * (3^2 / 1^3)
5 = k * (9 / 1)
(5 = 9k) / 9
5/9 = k
Thus, the constant of proportionality, k, is 5/9
Therefore, the relationship between P, Q, and Z is P = 5/9 * (Q^2 / Z^3)
II)
Determining the constant of proportionality, k:
Finding Z when P = 3 and Q = 5:
Now we can find Z when P = 3 and Q = 5 by substituting 3 for P, 5/9 for k, and 5 for Q:
3 = 5/9 * (5^2 / Z^3)
Simplifying on the right side gives us:
3 = 5/9 * (25 / Z^3)
Distributing the 5/9 to 25 / Z^3 gives us:
3 = (5 * 25) / (9 * Z^3)
3 = 125 / 9Z^3
Multiplying both sides by 9Z^3 give us:
(3 = 125 / 9Z^3) * 9Z^3
27Z^3 = 125
Dividing both sides by 27 gives us:
(27Z^2 = 125) / 27
Z^3 = 125/27
Taking the cube root of both sides gives us:
(Z^3 = 125 / 27)^(1/3)
Z = 1.666666667
Thus, Z is approximately 1.67 when P = 3 and Q = 5.