Worked problem — rectangle with given perimeter and area

Statement

The perimeter of a rectangle is $24$ cm and its area is $32$ cm². What is the length of its sides? Try to apply the four moves: understand, search for a route, carry out, review.

Solution

Understand. Data: perimeter $2p=24$, area $A=32$. Unknowns: base $x$ and height $y$. The statement becomes the system $\begin{cases} 2x+2y = 24\\ x\,y = 32 \end{cases}$ (see understanding an algebra problem).

Search for a route. Two equations, two unknowns: a well-posed problem. One may proceed by substitution, or with Vieta’s formulas (see searching for the route).

Carry out. Substituting $y=12-x$ into the second equation: $x(12-x)=32 \;\Longrightarrow\; x^2-12x+32=0 \;\Longrightarrow\; x=\frac{12\pm 4}{2},$ that is $x=8$ or $x=4$, with $y=4$ or $y=8$. The sides measure $\boxed{4\ \text{cm}}$ and $\boxed{8\ \text{cm}}$.

Review. Check: $2(4+8)=24$ ✓ and $4\cdot 8=32$ ✓. Extension: with perimeter $24$, the equation $x^2-12x+A=0$ has real solutions only if $A\le 36$; the greatest area is $36$, attained by the square with side $6$ (see reviewing the problem).

Links

Topics: Working method
Concepts: Data and unknowns · Problem solving
Skills: Modelling · Solving systems
Exercise type: Modelling problem