If y varies inversely as x2, how does x vary with y?

Analyze the question: y varies inversely as x², and we want to know how x relates to y.

Concept: Inverse variation means y = k/x², where k is a constant.

Solution:

1. Start with y = k/x²
2. Multiply both sides by x²: yx² = k
3. Divide by y: x² = k/y
4. Take the square root of both sides: x = √(k/y) = √k · (1/√y)

Since √k is just another constant, x = K/√y, where K = √k.

This means x varies inversely as √y.

Check the options:

• A. x inversely as y²: incorrect, since the power is ½, not 2.
• B. x inversely as √y: correct, matching x = K/√y.
• C. x directly as y²: incorrect.
• D. x directly as √y: incorrect, because the variation is inverse, not direct.
• E. x directly as y: incorrect.

Quick check: Let x = 2, so y = k/4. Take k = 4, giving y = 1. Now let x = 4, so y = 4/16 = 0.25. When x doubles, y falls to ¼ of its value, so √y halves, which means x is proportional to 1/√y. ✓

Exam tip: When you swap the subject in a variation statement, apply the opposite power to both sides. If y ∝ 1/x², then x ∝ 1/y^(1/2). The "inverse" stays inverse, and the power becomes its reciprocal.