Inverse variation: solve a practical problem

State the modelling assumptions

The tile calculations use an ideal area model. They ignore gaps, cutting and wastage. An area-only tile count is a model estimate; an actual layout can need more tiles

1. 1Fixed area The area of a square tile is its side length squared, so the area-only count varies inversely with that area

Which is a valid method for solving a stated variation problem?

Solving variation problems

A variation problem gives a relationship and a known pair of values. Find the positive constant k from that pair, then use the completed rule to substitute or solve for the requested value

1. 1Choose the rule Direct variation uses multiplication, while inverse variation uses a reciprocal \(\displaystyle y \propto x^{2} \Rightarrow y = kx^{2},\quad y \propto \frac{1}{x^{2}} \Rightarrow y = \frac{k}{x^{2}}\)
2. 2Find direct k For direct variation, divide the known output by the powered input \(\displaystyle y = kx^{2} \Rightarrow k = \frac{y}{x^{2}} = \frac{72}{3^{2}} = 8\)
3. 3Find inverse k For inverse variation, multiply the known output by the powered input \(\displaystyle y = \frac{k}{x^{2}} \Rightarrow k = x^{2}y = 5^{2} \times 6 = 150\)
4. 4Use the completed rule Substitute the target input, or undo the operations when the target input is unknown \(\displaystyle y = \frac{150}{5^{2}} = 6\)

Model and known pair

\(\displaystyle y=\frac{k}{x^2},\quad(x,y)=(4,18)\)

An ideal area model uses square tiles of side x cm. Find k in the inverse-square rule using the given tile-equivalent count and side length, ignoring gaps, cutting and wastage

Area-only model

\(\displaystyle y=\frac{288}{x^2},\quad x=6\)

The ideal area model gives tile-equivalent count y for square tiles of side x cm. Find the count for the given side length

An ideal area model gives 18 tile equivalents for square tiles of side 4 cm. Which calculation gives the estimate for side 6 cm, ignoring gaps and wastage?

Data or rule

\(\displaystyle N=\frac{500}{9^{2}}\)

An ideal area estimate covers 500 cm² using square pieces of side 9 cm. What is the minimum whole number by area, ignoring gaps and wastage?

An area of 500 cm² is to be covered by square pieces of side 9 cm, ignoring gaps and wastage. Why would rounding the area-only estimate to the nearest whole piece give too few pieces?

Data or rule

\(\displaystyle pt=k,\quad(p,t)=(4,6)\)

Equal-output pumps are modelled as sharing a fixed job with no delays. Four pumps take six hours. Predict the time for eight pumps

Four equal-output pumps take six hours to complete a fixed job. Eight pumps are predicted to take three hours. What would make this prediction unreliable?