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### Isosceles Triangles Poster

Isosceles Triangles Poster - February 2005

# Cuboid Challenge Poster

##### Age 11 to 14Challenge Level

The largest volume you can make is when $x = 3\frac13$,
which gives a volume of $592\frac{16}{27}$ cm$^2$.

Why?

For any value of $x$, the other sides are each $20-2x$ and then the volume is $x\times(20-2x)\times(20-2x)$

Trying out values of $x$, a spreadsheet program like Excel is useful

 $x$ $20-2x$ volume $x$ $20-2x$ volume $x$ $20-2x$ volume 2 16 512 2.5 15 562.5 3.1 13.8 590.36 3 14 588 3 14 588 3.2 13.6 591.87 4 12 576 3.5 13 591.5 3.3 13.4 592.55 5 10 500 3.6 12.8 589.82 3.4 13.2 592.42

You can keep going to get really close to 3.333333...

 $x$ $20-2x$ volume $x$ $20-2x$ volume 3.31 13.38 592.571 3.331 13.338 592.59237 3.32 13.36 592.585 3.332 13.336 592.59252 3.33 13.34 592.592 3.333 13.334 592.59259 3.34 13.32 592.591 3.334 13.332 592.59257

Using a cubic equation, the volume is $x(20-2x)^2$

which expands to $4x^3 - 80x^2 + 400x$

Differentiate to find the stationary points:

$\frac{\text{d}V}{\text{d}x} = 12x^2 - 160x + 400$

\begin{align} 12x^2-160x+400 &= 0\\ \Rightarrow 3x^2 - 40x + 100 &= 0\\ \Rightarrow 3x^2 - 30x -10x + 100 &= 0\\ \Rightarrow 3x(x-10) - 10(x-10)&=0\\ \Rightarrow (3x-10)(x-10)&=0\\ \Rightarrow 3x-10 = 0 \text{ or } x-10 = 0\end{align}

$x=10$ gives the minimum area of $0$, so $3x=10\Rightarrow x=\frac{10}3$ gives the maximum area.