Direct variation: write the relationship

The variable A varies directly as the square of r. Which equation states this?

Direct variation with a stated power

The symbol ∝ states a proportional relationship. Replace it by an equation with the same positive constant k for every input in the model

1. 1Keep the stated power For direct variation with a stated power, divide the known output by the powered input to find the constant \(\displaystyle y=kx^n,\qquad k=\frac{y}{x^n}\)
2. 2Use the completed rule Substitute for a forward value. Undo multiplication and then the power for a reverse value. Use the domain stated in the problem
3. 3Check the invariant At nonzero inputs, the output divided by the stated power stays fixed. A changing output-to-input ratio does not rule out direct variation with another stated power

Write the relationship: V varies directly as the cube of s

Write the relationship: T varies directly as the square root of L

If y varies directly as the square of positive x, what happens to y when x is tripled?

Write the whole equation before checking

On paper, write the equation for p directly proportional to the square root of q. Then find k if p is 15 when q is 25, and write the completed rule. This writing is self-reviewed

Model response: Write the whole equation before checking

The output p varies directly as the square root of q and equals 15 when q equals 25. Compare your solution with this model

1. 1Model response Substitute the known pair, solve for the constant, and state the rule for non-negative inputs \(\displaystyle \begin{aligned}p&=k\sqrt q\\15&=k\sqrt{25}=5k\\k&=3\\p&=3\sqrt q\quad(q\ge0)\end{aligned}\)

Data or rule

\(\displaystyle p=3\sqrt q,\quad q=49\)

Verify the completed rule at q=49.