Direct variation: find the constant and missing values

A dispenser delivers 18 litres in four minutes at a constant rate. How much does it deliver in ten minutes?

A tank starts with four litres and gains three litres per minute. Predict whether doubling the time from two to four minutes doubles the volume

The table uses y=kx+c. Compare c=4 with c=0, then change k. The columns show time x in minutes, volume y in litres, and y/x in L/min.

Which condition keeps y/x equal in every row for any displayed rate k?

A constant ratio and zero starting value

Direct proportion uses the same multiplicative scale factor for input and output. An added fixed amount breaks that property

1. 1Test matching nonzero inputs Divide each output by its matching nonzero input. Equal ratios support a direct model for the displayed data \(\displaystyle k=\frac{y}{x}\quad(x\ne0)\)
2. 2Check zero and the graph A direct rule has no fixed added amount. Its straight-line graph passes through the origin \(\displaystyle y=kx,\quad y(0)=0\)
3. 3Keep units consistent The constant has output units per input unit. Convert the input before applying a rate
4. 4Interpret the model Check the allowed inputs and whether the constant-rate assumption is reasonable. A pattern in recorded data does not prove the model outside those conditions

Data or rule

\(\displaystyle d=4.5\text{ m},\quad t=6\text{ s}\)

A conveyor moves a parcel 4.5 metres in six seconds at constant speed. Find its speed in metres per second

Data or rule

\(\displaystyle v=0.75\text{ m/s},\quad d=30\text{ m}\)

The conveyor moves at 0.75 m/s. Find the time needed to travel 30 metres

Which feature of this graph establishes direct proportion?

Why is this second straight-line model not direct proportion?

Data or rule

\(\displaystyle r=2.5\text{ L/min}\)

A hose delivers 2.5 litres per minute. Express the rate in litres per hour

Data or rule

\(\displaystyle r=2.5\text{ L/min},\quad t=90\text{ s}\)

Using that same hose rate, find the volume delivered in 90 seconds

The table includes (0,0), (2,7) and (4,14). How should the zero-input row be checked?

Data or rule

\(\displaystyle \begin{gathered}1\text{ cm represents }2.5\text{ m}\\l=3.6\text{ cm}\end{gathered}\)

A plan drawing uses 1 cm for 2.5 m of actual length. A wall measures 3.6 cm on the drawing. Find its actual length in metres

Data or rule

\(\displaystyle \begin{gathered}29\le V(2)\le31\text{ L}\\t=15\text{ min}\end{gathered}\)

A constant-flow test collects between 29 and 31 litres in exactly two minutes. Assuming the same flow continues, find the lower predicted volume for 15 minutes

Data or rule

\(\displaystyle \begin{gathered}29\le V(2)\le31\text{ L}\\t=15\text{ min}\end{gathered}\)

Use the same measured range of 29 to 31 litres in two minutes. Find the upper predicted volume for 15 minutes

A test gave 50 L in five minutes. A later test under nominally the same conditions gives 88 L in eight minutes. What should be concluded?

Data or rule

\(\displaystyle V(5)=55,\quad V(8)=88\)

After correcting the first measurement, the two readings are 55 L in five minutes and 88 L in eight minutes. Find the common rate

The corrected readings are 55 L in five minutes and 88 L in eight minutes. Which report is justified by these readings?

Extend the ratio to a stated power

The earlier direct-proportion work used y=kx. Now the input may be squared, cubed or square-rooted. Compute that quantity before finding the constant

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

\(\displaystyle y=kx^2,\quad (x,y)=(3,45)\)

Find the constant for this direct-square relationship

Data or rule

\(\displaystyle y=5x^2,\quad x=4\)

Use the completed direct-square rule at x=4.

Data or rule

\(\displaystyle 180=5x^2,\quad x>0\)

Recover the positive input when y=180.

Data or rule

\(\displaystyle y=kx^3,\quad(x,y)=(2,24)\)

Find k for a direct-cube relationship

Data or rule

\(\displaystyle y=k\sqrt{x},\quad(x,y)=(25,20)\)

Find k for a direct square-root relationship

Data or rule

\(\displaystyle 28=4\sqrt{x},\quad x\ge0\)

Recover the input for the square-root rule

The pairs (2,12), (3,27), (4,48) do not have a constant y/x ratio. What does stay constant?

Complete a powered-rule task

Write a full rule for z directly proportional to the cube of w, given that z is 40 when w is 2. Find z when w is 3, then recover positive w when z is 135. Check both answers in your rule

1. 1Write before continuing Write your solution before viewing the model on the next page

Model response: Complete a powered-rule task

Given that z is directly proportional to the cube of w and z is 40 when w is 2, compare your rule, calculations, constraints and explanation with this model

1. 1Model response Substitute the known pair to find the constant. Use the completed rule to find the output, then solve it backwards for the positive input \(\displaystyle \begin{gathered}40=8k,\quad k=5,\quad z=5w^3\\z=5(3^3)=135\\135=5w^3,\quad w^3=27,\quad w=3\end{gathered}\)