USA - Geometry Modelling with geometry

Design problems with geometric constraints

Q: What is a geometric design problem?

A problem that optimizes a shape subject to a constraint.

Q: What rectangle has the greatest area for a fixed perimeter?

A square gives the greatest area for a fixed perimeter.

Q: How do you set up a design problem?

Write the objective and the constraint, then reduce to one variable.

Q: How do you find the cost of a design?

Multiply the area or volume by the cost per unit.

20 practice questions 2 video lessons Theory + worked examples

Geometric Design Problems

California Geometry • Standard G-MG.3 • Modelling with Geometry

Geometric Design Problems is a topic in Modelling with Geometry in the California Common Core State Standards. It is aligned to Standard G-MG.3, which requires students to apply geometric methods to solve design problems, such as optimizing under constraints.

A design problem optimizes a shape — maximizing area, minimizing material, or lowering cost — subject to a geometric constraint.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

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Watch 2 video(s)

Cost of Fencing a Park - Perimeter Word Problems - Area & Perimeter 8 Watch
How many square tiles do I need for a 12x12 room? Watch

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:

\[\text{constraint: } 2x+2y=P,\qquad \text{objective: } A=xy\]

write the constraint linking the variables, then the objective to optimize

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 $100\ \text{ft}^2$.

a 10 by 10 square gives the maximum area of 100 square feet

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 $x=y=\sqrt{36}$.

$x=y$	$=$	$6\ \text{ft}$
$P$	$=$	$4(6)=24\ \text{ft}$

a 6 by 6 square uses the least fence, 24 feet

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 $r=4$.

$A$

$=$

$\pi(4)^2=16\pi\ \text{in}^2$

the largest circle has radius 4 and area 16 pi square inches

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.

$\text{area}$	$=$	$12\times15=180\ \text{ft}^2$
$\text{cost}$	$=$	$180\times 3=\$540$

the total cost is 540 dollars

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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\(A\)	\(=\)	\(20x-x^2\)
\(\text{vertex at } x\)	\(=\)	\(-\dfrac{20}{2(-1)}=10\)
\(y\)	\(=\)	\(20-10=10\)

\(x=y\)	\(=\)	\(6\ \text{ft}\)
\(P\)	\(=\)	\(4(6)=24\ \text{ft}\)

\(\text{area}\)	\(=\)	\(12\times15=180\ \text{ft}^2\)
\(\text{cost}\)	\(=\)	\(180\times 3=\$540\)

Design problems with geometric constraints

Geometric Design Problems

📖 Theory

How to solve a design problem

Common pitfalls

Frequently asked questions

Theory