Example — carrying out the rectangle problem

Example — Carrying out the rectangle problem

From the system $\begin{cases} 2x+2y=24 \\ xy=32 \end{cases}$, I isolate $y$ from the first equation: $y = 12 - x.$ I substitute into the second: $x\,(12-x) = 32 \;\Longrightarrow\; -x^2+12x-32=0 \;\Longrightarrow\; x^2-12x+32=0.$ I use the quadratic formula: $x = \frac{12\pm\sqrt{144-128}}{2} = \frac{12\pm 4}{2}.$ Hence $x=8$ or $x=4$. In the two cases, $y=12-x$ gives $y=4$ or $y=8$. The rectangle has sides $\boxed{4\ \text{cm}}$ and $\boxed{8\ \text{cm}}$ (the two solutions correspond to the same rectangle, seen from its base or from its height).

Links

Topics: Working method
Concepts: Write-up
Skills: Calculating · Solving equations · Solving systems