Construct a linear model from a description

Joining fee paid once

\(\displaystyle \$120\)

Fee per week

\(\displaystyle \$18\)

Which rule gives total cost C after w complete weeks?

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

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 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?

Report a model, its result and its limitations

A modelling report should let another person follow the method and judge whether to trust the result. Include the variables and units, assumptions, calculation, a check against observations, and a conclusion

1. 1Formulate Let d be the number of whole days and C the cost in dollars. Assume a one-off $40 booking fee and an unchanged $65 daily rate, with no other charges \(\displaystyle C=40+65d\)
2. 2Calculate and interpret Under those assumptions a six-day hire would cost $430 \(\displaystyle C(6)=40+65(6)=430\)
3. 3Evaluate The three-day invoice is $25 higher than the prediction. The current rule does not reproduce that observation \(\displaystyle \begin{gathered}C(3)=235\\260-235=25\end{gathered}\)
4. 4Conclude and communicate Treat $430 as a conditional prediction. Confirm the actual fee schedule and revise the model if needed before quoting a price. In your own report, distinguish a calculated result from a result supported by evidence

Run and record a linear-model investigation

Use actual graphing and symbolic technology, with a table or spreadsheet, for this hypothetical hire comparison. Plan A costs 30+12h dollars; plan B costs 60+6h dollars. Time h is continuous from 0 to 8 hours, and fractional hours are charged proportionally. Record the tool, actual inputs, settings, outputs and reasoning before opening the model. The full external work is self- or teacher-reviewed; the on-screen check concerns one stated feature

1. 1Formulate and choose functions Define variables and units, state both physical domains, and explain why a table, graph and symbolic equation solver answer different parts of this problem. If one tool lacks a required function, use another for that operation and record which you used. A preconfigured lesson graph or copied model is not your own tool run
2. 2Produce the table and family Generate a table for h=0,2,4,6,8 and graph both original models over the physical interval. Then replace plan B by B_k(h)=k+6h for k=36,60,84. Generate the three members in your own tool, keep A visible and label each curve. Predict which features change before inspecting the results
3. 3Choose and test the display Choose and record a window that shows all endpoints and every crossing in the physical interval. State the model domains and ranges separately from the viewing bounds. Deliberately crop the horizontal window to 0≤h≤4, then restore it. Explain why failing to see a crossing in that crop does not establish that none occurs in the allowed interval
4. 4Use symbolic computation and test a claim Use your symbolic tool to simplify A(h)−B_k(h) and solve A(h)=B_k(h) for h, including the three stated k-values. Test the claim that two different hourly rates guarantee an equal-cost booking in the allowed interval. Produce a technology example and a counterexample, then verify each by exact substitution and the domain condition. Do not round an inadmissible time into the interval
5. 5Compare notation and report Write the formal equations beside the exact expressions or commands you entered. Explain how your tool represents multiplication, brackets, equality or assignment, the parameter k and any domain restriction. Distinguish an exact symbolic result from a rounded graph reading. Report formulation, operations, independent checks and a practical conclusion, including the fixed-rate/no-extra-charges assumption

Review the outputs and the admissibility argument

For h=0,2,4,6,8, plan A gives 30,54,78,102,126 dollars. B with k=36 gives 36,48,60,72,84; k=60 gives 60,72,84,96,108; k=84 gives 84,96,108,120,132. Each B graph has gradient 6 dollars per hour; changing k changes only its fixed fee. The reference figure in the following comparison check shows the original k=60 pair, not a substitute for producing the full family

1. 1Domain, range and useful windows Every physical domain is [0,8]. The A range is [30,126]; the three B ranges are [36,84], [60,108] and [84,132]. For example, −0.5≤h≤8.5 and 0≤C≤140 displays the supported segments with margin. Restrict the plotted model to 0≤h≤8 rather than treating the margin as extra valid hire times. A horizontal crop to [0,4] hides the original crossing at h=5
2. 2Exact crossings and a counterexample For k=36 the crossing is (1,42); for k=60 it is (5,90). Both are admissible examples. For k=84, the unique algebraic crossing is (9,138), outside [0,8]. On the physical interval, A−B_84=6h−54≤−6, so A remains cheaper. Different hourly rates guarantee one intersection of the unrestricted lines, not an admissible booking
3. 3Verify independently At h=5, 30+12×5=90=60+6×5. At h=9, both unrestricted expressions with k=84 give 138, but the domain rejects that booking. Recompute selected table entries without the table formula and compare both endpoints. A graph or solver answer alone does not check units, assumptions or admissibility
4. 4Document the actual technology An input may use 30+12*h instead of the formal 30+12h, and (k-30)/6 must keep the numerator grouped. A symbolic solve operation should recover the crossing shown above; the exact command and parameter handling depend on the chosen tool. Record what you actually entered and obtained, including how the physical interval was enforced. Do not invent a universal command or report only this model response
5. 5Interpret the original comparison For k=60, A is cheaper below 5 hours, both cost $90 at 5 hours and B is cheaper above 5 hours up to 8. The result assumes the given continuous charging rules without rounding to whole hours, extra fees or a changed rate

This reference diagram shows the original pair A=30+12h and B=60+6h. What does its crossing mean?

Technology returns h=9 for equality of A(h)=30+12h and B(h)=84+6h. Hire is restricted to 0≤h≤8. Which conclusion is justified?