Distance in first 3 s: $s_3 = \dfrac{1}{2}(10)(3)^2 = 45\text{ m}$

Distance in last 1 s of total time $T$ (using $s_n = u + \dfrac{a}{2}(2n-1)$ with $u=0$):

$s_{\text{last}} = \dfrac{g}{2}(2T-1) = 5(2T-1)$

Setting equal: $5(2T-1) = 45 \Rightarrow 2T-1 = 9 \Rightarrow T = \mathbf{5\text{ s}}$

Plain cement concrete (PCC):

• Very high compressive strength (~20–50 MPa for standard mixes)
• Very low tensile strength (~1/10th of compressive = 2–5 MPa)
• This is why steel reinforcement is added (RCC) — steel handles tension, concrete handles compression.

Structural steel has high strength in both tension and compression. Timber has moderate strength in both. RCC is a composite with improved tensile properties.

Among single materials, PCC best demonstrates this extreme disparity.

For non-zero solution, $\Delta=0$:

$\begin{vmatrix}1&k&3\\3&k&-2\\2&4&-3\end{vmatrix}=1(−3k+8)−k(−9+4)+3(12−2k)=−3k+8+5k+36−6k=44−4k=0$

So $k=11$.

With $k=11$: from equations 1 and 2: $x+11y+3z=0$ and $3x+11y-2z=0$. Subtracting: $2x-5z=0\Rightarrow x=\frac{5z}{2}$.

Substituting back: $\frac{5z}{2}+11y+3z=0\Rightarrow 11y=-\frac{11z}{2}\Rightarrow y=-\frac{z}{2}$.

$\frac{xz}{y^2}=\frac{\frac{5z}{2}\cdot z}{\frac{z^2}{4}}=\frac{\frac{5z^2}{2}}{\frac{z^2}{4}}=\frac{5}{2}\times4=\mathbf{10}$

Let $\sin^{-1}x = \pi/2-\epsilon$ where $\epsilon\to0^+$ as $x\to1^-$. Then $x=\sin(\pi/2-\epsilon)=\cos\epsilon\approx1-\epsilon^2/2$, so $1-x\approx\epsilon^2/2$.

$\sqrt{\pi}-\sqrt{2(\pi/2-\epsilon)}=\sqrt{\pi}-\sqrt{\pi-2\epsilon}=\sqrt{\pi}\left(1-\sqrt{1-\frac{2\epsilon}{\pi}}\right)\approx\sqrt{\pi}\cdot\dfrac{\epsilon}{\pi}=\dfrac{\epsilon}{\sqrt{\pi}}$

$\sqrt{1-x}\approx\sqrt{\epsilon^2/2}=\epsilon/\sqrt{2}$

Limit $=\dfrac{\epsilon/\sqrt{\pi}}{\epsilon/\sqrt{2}}=\sqrt{\dfrac{2}{\pi}}$

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