Formulate and solve a quadratic problem

A service charges a $30 call-out fee plus $5 per item. Which model type fits the total charge C for n items?

A rectangular garden is 4 m longer than it is wide. Change its width by 1 m at a time. Compare the increases in perimeter and area

What explains the different growth patterns?

A complete modelling cycle

Define the problem and variables, choose and justify a relationship, solve it, interpret the answer, then evaluate and communicate the result

1. 1Choose Use how quantities are related, not the topic title, to decide whether a model is linear, quadratic or another simple variation
2. 2Check State units and allowed inputs. Compare predictions with the original conditions or with observations
3. 3Report Give the rule, method, result, assumptions and limitations

For a rectangle 4 m longer than its width w, which equation models an area of 60 m²?

Width

\(\displaystyle w=6\)

Length rule

\(\displaystyle l=w+4\)

The rectangle has width 6 m. What is its length?

Use the length rule with the stated width

Width

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

Length

\(\displaystyle 10\text{ m}\)

For the same 6 m by 10 m rectangle, calculate its perimeter

Add two copies of each side length

If both lengths of a rectangle double, what happens to its area?

Counts form a product

A display has r rows and r+2 objects in each row. The total is the product of the two counts. The number of rows must be a positive whole number

A display has r rows, with r+2 objects in each row and 48 objects altogether. Which equation models the total?

Roots

\(\displaystyle r=-8\text{ or }6\)

A display has r rows, with r+2 objects in each row and 48 objects altogether. The model has roots −8 and 6. How many rows fit the display?

Choose the root that can be a positive row count

An equation for the number of rows in a display has a root of −8. Why is that root rejected?

A perimeter determines the other side

A rectangle has perimeter 30 m and area 54 m². If its shorter side is x, the other side is 15−x. Solving gives x=6 or x=9; the shorter-side condition keeps x=6.

Half-perimeter

\(\displaystyle 15\text{ m}\)

Shorter side

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

The shorter side is 6 m and the half-perimeter is 15 m. Find the other side

Subtract the shorter side from the half-perimeter

For a rectangle with sides 6 m and 9 m, x was defined as the shorter side. Both values are positive roots of the area equation. Why use only 6 for x?

For equally spaced inputs 0,1,2,3, outputs are 5,8,11,14. Which simple model matches these values?

For widths 0,1,2,3, the rule A=w(w+4) gives 0,5,12,21. Which pattern supports a quadratic rule?

A few measurements are evidence, not proof

Several rules can fit a short table. Justify a model using the relationship and assumptions as well as differences. Measured data may be noisy; compare predictions with further observations

Simple interest is linear

For $1,200 at 5% simple interest per year, interest is calculated on the original principal. Each year adds $60. Assume a fixed rate, no fees and no deposits or withdrawals. Let t be time in years, I the interest and B the balance, both in dollars

Balance rule

\(\displaystyle B=1{,}200+60t\)

Years

\(\displaystyle t=3\)

Under those assumptions, what is the balance after 3 years?

Substitute the number of years into the simple-interest balance rule

Other simple variations

An idealised job takes 24 worker-hours. If n equally productive workers share it perfectly, the duration in hours is given by the displayed rule. The fixed-work assumption gives a reciprocal model, not a linear or quadratic one

An idealised job takes 24 worker-hours, shared perfectly by equally productive workers. Which prediction follows?

Evaluate a prediction

A measured garden near width 3 m has area 20 m². The ideal length w+4 predicts 21 m². The 1 m² difference calls for checking measurements and whether the shape is truly rectangular; it does not automatically invalidate every use of the model

An ideal rectangle model predicts a garden area of 21 m², while the measured area is 20 m². Which report communicates this result responsibly?

Your modelling investigation

Choose a rectangular surface you can measure safely. Investigate its perimeter, area or material cost as a dimension changes

1. 1Define and choose Define width and length with units and a realistic domain. State assumptions and justify a linear, quadratic or other appropriate relationship
2. 2Predict and check Calculate two predictions. Compare one with a measurement and report the difference. Include a table or graph
3. 3Report and review Explain the method, contextual result and a limitation. Ask a teacher or peer to review your reasoning. Automated practice answers do not certify this investigation