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P Limited produces chairs at $70 each. It receives a special order for 1,000 chairs with padded seats. This requires $6,000 in extra materials, 500 extra labour hours at $15/hour, and $2,000 extra overheads. What is the total cost of this batch?
1. Base cost = 1,000 $\times$ $70 = $70,000.
2. Extra materials = $6,000.
3. Extra labour = 500 $\times$ $15 = $7,500.
4. Extra overheads = $2,000.
Total = $70,000 + $6,000 + $7,500 + $2,000 = $85,500.
Evaluate: $\displaystyle\lim_{x\to1^-}\dfrac{\sqrt{\pi}-\sqrt{2\sin^{-1}x}}{\sqrt{1-x}}$
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 $=\dfrac{\epsilon/\sqrt{\pi}}{\epsilon/\sqrt{2}}=\sqrt{\dfrac{2}{\pi}}$
- $V^{2+}$: $[Ar] 3d^{3}$ (3 unpaired)
- $Cr^{2+}$: $[Ar] 3d^{4}$ (4 unpaired)
- $Mn^{2+}$: $[Ar] 3d^{5}$ (5 unpaired)
- $Fe^{2+}$: $[Ar] 3d^{6}$ (4 unpaired)
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