Suppose X varies jointly as the square of Y and cube root of Z and inversely as the fifth power of W. If X = 6.75 when Y = 8, Z = 27, and W = 4, find X when Y = 16, Z = 163 and W = 12. Round only the final answer to 4 decimal places.
X = k * Y^2 * Z^(1/3) / W^5
6.75 = k * 8^2 * 27^(1/3) / 4^5
solve for k (the constant of variation)
then use it with the 2nd set of values to find X
To find X when Y = 16, Z = 163, and W = 12, we can use the concept of joint variation and inverse variation.
Let's denote the constant of variation as k.
According to the problem, X varies jointly as the square of Y and the cube root of Z and inversely as the fifth power of W. We can write this relationship as:
X = k * (Y^2) * (Z^(1/3)) / (W^5)
To find the value of k, we can use the given information when X = 6.75, Y = 8, Z = 27, and W = 4:
6.75 = k * (8^2) * (27^(1/3)) / (4^5)
Simplifying the equation:
6.75 = k * 64 * (3^(1/3)) / 1024
To find k, divide both sides of the equation by (64 * (3^(1/3)) / 1024):
k = 6.75 / (64 * (3^(1/3)) / 1024)
Simplifying the expression for k:
k ≈ 0.5039
Now that we have the value of k, we can find X when Y = 16, Z = 163, and W = 12:
X = 0.5039 * (16^2) * (163^(1/3)) / (12^5)
Calculating each component:
X ≈ 0.5039 * 256 * (163^(1/3)) / 248,832
Simplifying:
X ≈ 41.2928 * (163^(1/3))
Rounding to four decimal places, X is approximately:
X ≈ 41.2928 * 5.3534
X ≈ 221.1555
Therefore, X is approximately 221.1555 when Y = 16, Z = 163, and W = 12 (rounded to four decimal places).
To find X when Y = 16, Z = 163, and W = 12, you need to apply the concept of joint variation and inverse variation.
Joint variation is a relationship in which a variable varies directly or jointly with two or more variables. In this case, X varies jointly as the square of Y and the cube root of Z.
Inverse variation is a relationship in which a variable varies inversely with another variable raised to a power. In this case, X varies inversely as the fifth power of W.
Let's break down the given information and solve for X.
We're given:
X varies jointly as the square of Y and the cube root of Z.
X varies inversely as the fifth power of W.
So, we can write the equation as:
X = k * Y^2 * Z^(1/3) / W^5
To determine the value of the constant of variation, k, we can use the given information when X = 6.75, Y = 8, Z = 27, and W = 4.
Substituting these values into the equation:
6.75 = k * 8^2 * 27^(1/3) / 4^5
Simplifying:
6.75 = k * 64 * 3 / 1024
6.75 = k * 192 / 1024
6.75 = k * 0.1875
To find k, divide both sides of the equation by 0.1875:
k = 6.75 / 0.1875
k ≈ 36
Now that we have the value of k, we can find X when Y = 16, Z = 163, and W = 12.
Substituting these values into the equation:
X = 36 * 16^2 * 163^(1/3) / 12^5
Simplifying:
X ≈ 36 * 256 * 5.1648 / 248832
X ≈ 47.2126
Therefore, when Y = 16, Z = 163, and W = 12, X is approximately equal to 47.2126 rounded to 4 decimal places.