Direct variation: solve a practical problem

Model a direct-variation situation

The problem must state the relationship or give evidence for it. Keep units consistent, use the stated power, find k from one pair and substitute into the completed rule

1. 1Keep the stated power For y directly proportional to xⁿ, divide the known output by the powered input to find k. \(\displaystyle y=kx^n,\qquad k=y/x^n\)
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 The ratio y/xⁿ is fixed at nonzero x. A changing y/x ratio does not rule out direct variation with a different stated power

Data or rule

\(\displaystyle C=ks^2,\quad(s,C)=(3,63)\)

Painting cost is modelled as directly proportional to the area of a square panel, with no fixed fee. A panel of side three metres costs $63. Find the rate per square metre

Data or rule

\(\displaystyle C=7s^2,\quad s=5\)

Predict the painting cost for a panel of side five metres under the same model

Data or rule

\(\displaystyle 252=7s^2,\quad s>0\)

Find the positive side length for a painting cost of $252

A painting cost has a part directly proportional to area. A nonzero fixed call-out fee is added. Which conclusion follows?

Data or rule

\(\displaystyle y=k\sqrt{x},\quad y(9)=6\)

A stated model has y directly proportional to √x. The known pair is (9,6). Find y at x=25.

Data or rule

\(\displaystyle 14=2\sqrt{x},\quad x\ge0\)

For the completed square-root model, recover x from y=14.

Write a complete practical solution

A model gives the material cost of a square sheet as directly proportional to its area. A side length of 1.5 m costs $27. Write the rule, predict the cost for side 2 m, and describe two assumptions

1. 1Write before continuing Write your complete solution independently. The next screen provides a model for self-review; the full written response is not automatically marked

Model response: Write a complete practical solution

A square sheet of side length 1.5 m costs $27 in materials, with cost directly proportional to area. Find the rule and the cost for side length 2 m, then state two assumptions. Compare your work with this model

1. 1Model response C=ks² and 27=2.25k give k=$12/m². Hence C=12s² and C(2)=$48. The model assumes a constant price per unit area, no fixed charge and no extra waste cost. It does not prove those conditions hold for another supplier