USA - Geometry
Modelling with geometry
Design problems with geometric constraints
20 practice questions
2 video lessons
Theory + worked examples
Theory
A design problem asks for the best shape or size under a constraint:
- Objective: what to maximize or minimize (area, material, cost).
- Constraint: a fixed quantity (perimeter, area, budget).
- Feasibility: the answer must make physical sense.
Useful facts: for a fixed perimeter the rectangle of greatest area is a square; for a fixed area the square has the least perimeter.
Maximize area for a fixed perimeter: the best rectangle is a square.
The design-problem process.
A typical setup:
\[\text{constraint: } 2x+2y=P,\qquad \text{objective: } A=xy\]
Use the constraint to eliminate a variable, then optimize the resulting single-variable expression.
How to solve a design problem
- State the objective (what to maximize or minimize).
- Write the constraint as an equation.
- Substitute to get one variable, then optimize.
- Check units and that the answer is physically reasonable.
Example 1 — Maximize area, fixed perimeter
A rectangular pen uses \(40\) ft of fence. What dimensions give the greatest area?
Solution
With perimeter \(2x+2y=40\), we have \(y=20-x\), so area \(A=x(20-x)\).
| \(A\) | \(=\) | \(20x-x^2\) |
| \(\text{vertex at } x\) | \(=\) | \(-\dfrac{20}{2(-1)}=10\) |
| \(y\) | \(=\) | \(20-10=10\) |
A \(10\times10\) square, area \(100\ \text{ft}^2\).
Example 2 — Minimize fence, fixed area
A rectangular garden must have area \(36\ \text{ft}^2\). What dimensions use the least fencing?
Solution
For a fixed area the perimeter is smallest when the rectangle is a square, so \(x=y=\sqrt{36}\).
| \(x=y\) | \(=\) | \(6\ \text{ft}\) |
| \(P\) | \(=\) | \(4(6)=24\ \text{ft}\) |
Example 3 — Fit within a constraint
What is the largest circle that fits inside an \(8\)-inch square, and its area (leave \(\pi\))?
Solution
The circle's diameter equals the side, so \(r=4\).
| \(A\) | \(=\) | \(\pi(4)^2=16\pi\ \text{in}^2\) |
Example 4 — Cost of a design
Tiling a \(12\times15\) ft floor costs \$3 per square foot. Find the total cost.
Solution
Multiply the area by the unit cost.
| \(\text{area}\) | \(=\) | \(12\times15=180\ \text{ft}^2\) |
| \(\text{cost}\) | \(=\) | \(180\times 3=\$540\) |
Common pitfalls
Don't ignore the constraint; the objective alone has no finite best answer.
Check feasibility: lengths must be positive and fit the situation.
Match units for cost problems — cost per unit area times area.
Frequently asked questions
What is a geometric design problem?
A problem that optimizes a shape (area, material, or cost) subject to a constraint like a fixed perimeter or budget.
What rectangle has the greatest area for a fixed perimeter?
A square — equal sides give the maximum area.
How do you set up a design problem?
Write the objective to optimize and the constraint equation, then use the constraint to reduce to one variable.
How do you find the cost of a design?
Multiply the area (or volume) by the cost per unit.
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