Construct a linear model from a description

Observed purchase

\(\displaystyle 3\text{ m costs }\$7.50\)

Variables

\(\displaystyle C\text{ in dollars},\quad l\text{ in metres}\)

Fabric is sold at a constant price per metre with no fixed charge. Which rule gives its cost from its length?

The machines agree at the starting input. Double the input, then compare each machine's outputs in the Start and Now columns

What happened when you doubled the input?

Recognise direct proportion

In direct proportion, multiplying the input by a factor multiplies the output by the same factor. A constant rate and a zero starting value give this model

1. 1Find the rate Divide a nonzero input's corresponding output by that input. Attach output units per input unit \(\displaystyle k=\frac{y}{x}\quad(x\ne0)\)
2. 2Check the starting value A nonzero fixed additional amount prevents direct proportion, even when the graph is a straight line \(\displaystyle y(0)=0\)
3. 3Use the model in either direction Multiply by the rate to find the output. Divide a target output by a nonzero rate to find the input \(\displaystyle \begin{aligned}y&=kx\\x&=\frac{y}{k}\quad(k\ne0)\end{aligned}\)
4. 4Respect the situation Use consistent units and check the allowed inputs. For a positive rate, divide an output limit by the rate to find the input limit. Counted objects need whole-number inputs; measured lengths may allow fractional inputs

Observed purchase

\(\displaystyle 2.4\text{ m costs }\$9.60\)

Required length

\(\displaystyle 3.5\text{ m}\)

Ribbon has a constant price per metre and no fixed charge. Find the cost of the required length

Find the price of one metre, then multiply by the required length

Observed purchase

\(\displaystyle 6\text{ patches cost }\$15\)

Budget

\(\displaystyle \$22.50\)

Identical canvas patches have a constant price with no fixed charge. What is the greatest number of patches that can be bought within the budget?

Find the price of one patch and divide the budget by that price

Rule

\(\displaystyle y=2x+4\)

A student says a straight-line rule must show direct proportion. What is wrong with this claim for the displayed rule?

Joining fee paid once

\(\displaystyle \$120\)

Fee per week

\(\displaystyle \$18\)

Which rule gives total cost C after w complete weeks?

A battery starts at 80% charge and loses 8 percentage points per hour. Predict its charge after two hours

Change the hourly drain, then the starting charge. Compare the zero-hour row with later rows. The drain is measured in percentage points, not a percentage of the remaining charge

What does changing only the drain rate leave unchanged?

Starting value, rate and permitted inputs

Name the input and output, keep their units consistent, and use the model only for its stated conditions

1. 1Construct the rule The constant gives the output at zero input. The signed coefficient gives the change per input unit \(\displaystyle y=a+bx\)
2. 2Use the sign An increasing output has a positive rate. A decreasing output has a negative rate \(\displaystyle V=80-8t\)
3. 3Check input units Convert to the unit used by the model before substituting \(\displaystyle 2\text{ hours}=120\text{ minutes}\)
4. 4Check the domain An interval limits where the model is supported. Some contexts also require whole-number inputs \(\displaystyle 0\le t\le10\)

Model or data

\(\displaystyle C(2)=94,\quad C(5)=166\)

A service charges a fixed booking fee plus an unchanged hourly rate. Two hours costs $94 and five hours costs $166. Find the hourly rate

Model or data

\(\displaystyle C= a+24h,\quad C(2)=94\)

Use the same service data and the rate of $24 per hour. Find the one-off booking fee

Which observation supports a linear model for equally spaced inputs?

Model or data

\(\displaystyle C=46+24h\)

The service uses C=46+24h, where h is time in hours and C is dollars. Find the cost of a 90-minute booking. Fractional hours are charged proportionally

Booking fee paid once

\(\displaystyle \$40\)

Daily charge

\(\displaystyle \$65\)

Hire length

\(\displaystyle 6\text{ days}\)

Find the cost of a six-day hire. Only whole days are charged

Add the one-off fee to the daily charge multiplied by the number of days

Booking fee paid once

\(\displaystyle \$40\)

Daily charge

\(\displaystyle \$65\)

Budget

\(\displaystyle \$650\)

Find the largest whole number of days the budget covers

Remove the one-off fee, divide the remaining budget by the daily charge, then check whole-day costs

Check the two hire calculations

A budget boundary need not be a whole number. Test the largest permitted whole input and the next one

1. 1Predict a cost Substitute six days \(\displaystyle \begin{gathered}C(6)=40+65(6)\\=40+390=430\end{gathered}\)
2. 2Find the budget boundary The cost must not exceed the budget \(\displaystyle \begin{aligned}40+65d&\le650\\65d&\le610\\d&\le\frac{122}{13}\approx9.38\end{aligned}\)
3. 3Apply the whole-day condition Nine days fits. Ten days does not \(\displaystyle \begin{gathered}C(9)=625\le650\\C(10)=690>650\end{gathered}\)

Model or data

\(\displaystyle \begin{gathered}V=80-8t\\0\le t\le10\end{gathered}\)

A battery model gives charge V as a percentage after t hours. At what time does it reach 20%? Use the stated interval

Model for charge percentage

\(\displaystyle V=80-8t\)

Permitted time in hours

\(\displaystyle 0\le t\le10\)

Proposed time

\(\displaystyle t=8.5\)

Does the model support a prediction at the proposed time?

Starting charge

\(\displaystyle 80\%\)

Drain per hour

\(\displaystyle 8\text{ percentage points}\)

Where is the first error in this decreasing-charge model?

Proposed cost model in dollars

\(\displaystyle C=40+65d\)

Whole days hired

\(\displaystyle d=3\)

Observed invoice

\(\displaystyle \$260\)

Check the hire model against a new invoice. How many dollars higher is the invoice than the predicted cost?

Predict the three-day charge, then subtract that prediction from the observed invoice

The model predicts $235 for three days, but the invoice is $260. Which conclusion is supported?

The original hire rule was C=40+65d. Two invoices under the same charging conditions are $260 for three days and $390 for five days. Assuming a fixed fee and constant daily rate, which revised rule fits both?

Model or data

\(\displaystyle C=65+65d\)

Using the revised rule under the same charging conditions, predict the cost of six days

Explain a revision and its limits

The original fixed fee was $40. The invoices were $260 for three days and $390 for five days. These observations support a revised fixed-fee model, but do not prove it applies to every future booking

1. 1Formulate and check State whole days d and cost C in dollars. Both observed invoices fit a $65 fixed fee and $65 daily rate \(\displaystyle C=65+65d\)
2. 2Interpret A six-day booking is predicted to cost $455 under the same charging conditions
3. 3Evaluate and report Identify the changed assumption: the fixed fee was $65 rather than $40. Confirm that the invoices contain no other charges and that the rate stays constant over the intended hire length. State those limitations alongside the result