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C1.4 · Solve design problems under constraints and state assumptions

Learn to solve design problems under constraints and state assumptions through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Geometry and Trigonometry

MBF3C study topic C1.4: solve design problems and state assumptions

A design problem asks you to choose or plan something while meeting stated conditions. Those conditions are constraints. For example, a garden may need a certain area but have only a fixed amount of fencing. A poster may need a minimum area and must stay within a budget. Solving the problem means using the information to find a design that works, then checking it carefully. The answer may also depend on assumptions: reasonable details treated as true because the problem does not specify them. This lesson uses familiar measurement, arithmetic, variables, and equations to make and explain design decisions.

What you will learn

Identify a design problem’s goal, constraints, and given information.
Choose variables and represent relationships between design quantities.
Test a possible design against every constraint.
Explain a solution and state assumptions that affect it.

1. Read the design brief

Start by finding the goal, the constraints, and the information supplied. The goal is what you are trying to accomplish. A constraint is a limit or requirement that the design must satisfy. Given information might include measurements, prices, or rules. For example, “at least 50 square metres” is a minimum area requirement, while “no more than CAD 28” is a maximum cost.

Before calculating, put the quantities in a form you can use. A variable is a letter that stands for a quantity that may change. For a rectangular design, you could use

w

for width and

l

for length. A relationship is a rule connecting quantities. A rectangle’s area is found by multiplying its length by its width. If fencing goes around all four sides, the total fencing depends on both measurements.

Check that units are consistent. For instance, convert centimetres to metres before combining a length in centimetres with a length in metres. Also check whether a proposed value makes sense in the situation. A negative width cannot describe a real poster or garden.

A=lw

Goal: what the design should accomplish.
Constraint: a limit or requirement the design must obey.
Use variables to represent quantities that can change.
Keep units consistent and reject impossible measurements.

2. Build and use a simple model

A model is a simplified mathematical description of a real situation. Choose variables for the measurements or choices you need to decide. Then use the given conditions to connect them. If a rectangular garden has a fixed amount of fencing on three sides, the two widths and one length must add to that amount.

You can test a possible design by substituting its measurements into the relationships. Substitution means replacing a variable with a chosen value. Calculate the result, then compare it with the requirement. A table can help if you want to compare several choices for one measurement. A graph or suitable technology can also help show how choices relate. Whatever representation you use, it must follow the conditions in the design brief.

An assumption is a reasonable detail you treat as true when the problem does not provide it. For example, you might assume that a wall can serve as one side of a garden, or that a printing price has no setup fee. Assumptions are not facts supplied by the problem. State them so another person can see what your answer depends on.

2w+l=30

Use the conditions in the brief to connect variables.
Test candidate measurements by calculating what they produce.
Tables and graphs can help compare possible designs.
State missing details you have assumed.

3. Check all constraints and explain the result

A candidate design is one possible set of measurements or choices. Check it against every constraint, not just the main goal. A poster can have enough area and still cost too much. A garden can have enough space and still require more fencing than is available.

If the problem asks for the best design, identify what “best” means in that situation. It might mean the lowest cost, or it might mean any design that satisfies all stated requirements. Compare only designs that meet the constraints. Do not claim that a design is best unless the information and comparisons support that claim.

A complete answer gives the proposed design, the checks, and the assumptions. Include units with measurements and costs. If a design is approximate, explain why, such as when materials are only available in fixed lengths. Clear checks make it possible for someone else to follow your decision.

Check each constraint separately.
Explain what “best” means before comparing designs.
Include units and show the important checks.
State assumptions that may affect the answer.

4. A practical solving routine

A reliable routine is to describe the goal and list the constraints first. Next, name the quantities you can choose and represent how they are connected. Then choose a candidate design and calculate its measurements, area, or cost. Compare the results with every requirement. If one condition fails, revise the design and test again.

For example, if a design must have at least a target area and no more than a maximum width, a good first choice may be the largest allowed width. You can then calculate the length needed for the target area and check any other limits. This is a practical choice to test, not a guarantee that it solves every version of the problem.

Finally, write a short conclusion in context. Say what to build or choose, why it satisfies the constraints, and what assumptions you made. This connects the calculations back to the original design.

Translate the brief into quantities and requirements.
Test a specific design and revise it if necessary.
Finish with a conclusion that includes checks and assumptions.

Worked example

A rectangular garden beside a wall

A community group has 30 m of fencing for a rectangular garden. One side lies along a wall, so fencing is needed on the other three sides. The garden must have an area of at least 50 m². Find one possible design and state assumptions.

Choose variables
Let $w$ be each width perpendicular to the wall, and let $l$ be the length parallel to it. The fence covers two widths and one length, so those three sides must use the available 30 m.
$2w+l=30$
Choose a width
Try a width of $5$ m. The two widths use $10$ m of fencing. Subtracting that amount from the total leaves the length of the third fenced side.
$l=30-2(5)=20\text{ m}$
Check the area
Multiply the length by the width to find the rectangular area. The result is greater than the required minimum of $50$ m².
$A=lw=20(5)=100\text{ m}^2$
State assumptions
Assume the wall forms a straight side, the ground is level, and no fencing is needed for a gate or supports. These details were not specified, so they should be stated.

Answer: One possible design is 20 m along the wall and 5 m perpendicular to it. Its area is 100 m², and it uses 30 m of fencing on the other three sides.

Check: The fenced sides total 5 + 5 + 20 = 30 m. The area, 100 m², is at least 50 m².

Worked example

A poster within size and budget limits

A school club is choosing a rectangular poster. Its area must be at least 600 cm², its width must be no more than 20 cm, and printing costs CAD 0.04 per square centimetre. The budget is CAD 28. Find a design that meets the requirements and state assumptions.

Name the measurements
Let $w$ be the width in centimetres and $l$ be the length in centimetres. The area depends on both measurements, so use the rectangle area relationship.
$A=lw$
Choose a width and find a length
Choose the maximum permitted width, $20$ cm. For an area of $600$ cm², divide the target area by the width. This gives a length that reaches the minimum area exactly.
$l=600\div20=30\text{ cm}$
Check the area and cost
The dimensions give the required area. Multiply that area by the printing rate to calculate the cost. The result is less than the CAD 28 budget.
$A=20(30)=600\text{ cm}^2,\quad C=600(0.04)=24$
State assumptions
Assume the printer charges exactly CAD 0.04 per square centimetre and adds no setup fee, tax, or extra charge for these dimensions. If the price rules differ, the cost check may change.

Answer: Choose a poster that is 20 cm wide and 30 cm long. Its area is 600 cm² and its printing cost is CAD 24.

Check: The width is at the 20 cm maximum, the area meets the 600 cm² minimum, and CAD 24 is within the CAD 28 budget.

Common mistakes and how to avoid them

Checking only the main goal, such as area, and forgetting another limit.

Correction: List each constraint and check the proposed design against every one.

Combining measurements that use different units.

Correction: Convert to consistent units before calculating, then include units in the result.

Treating a missing detail as guaranteed without saying so.

Correction: State reasonable assumptions, such as no setup fee or no gate opening, and explain what they affect.

Lesson summary

Identify the goal, constraints, and given information before calculating.
Choose variables and use the brief to connect the quantities.
Test a candidate design against every constraint.
Communicate the design with units and state relevant assumptions.

Check your understanding

Question 1

A rectangular sign must have an area of at least 240 cm². Its width is 12 cm. Which length is the shortest listed choice that meets the area requirement?

18 cm
19 cm
21 cm
24 cm

Show answer and explanation

21 cm

At 18 cm, the area is 216 cm²; at 19 cm, it is 228 cm². Both are below 240 cm². At 21 cm, the area is 252 cm², so it meets the requirement. Since 21 cm is shorter than the other qualifying listed choice, it is the shortest listed choice that works.

Question 2

A design problem gives a maximum budget but does not say whether delivery is included. What should you do when using a model?

Ignore the budget because one cost is unknown.
State a reasonable assumption about delivery and explain it.
Treat delivery as free without mentioning it.
Add an invented delivery price to the given budget.

Show answer and explanation

State a reasonable assumption about delivery and explain it.

The missing detail should be handled as an explicit assumption. That lets the reader see what the result depends on.

Key terms

Constraint: A limit or requirement that a design must satisfy.
Variable: A letter used to represent a quantity that can change.
Model: A simplified mathematical description of a real situation.
Assumption: A reasonable detail treated as true when the problem does not provide it.
Candidate design: One possible set of choices being tested against the constraints.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation C1.4. It is a study resource, not an official curriculum publication.

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