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Spot size $b \approx \text{geometric spread} + \text{diffraction spread} = a + \frac{L\lambda}{a}$.
To minimize $b$, differentiate with respect to $a$ and set to zero:
$\frac{db}{da} = 1 - \frac{L\lambda}{a^2} = 0 \implies a = \sqrt{\lambda L}$.
Substituting back: $b_{min} = \sqrt{\lambda L} + \frac{L\lambda}{\sqrt{\lambda L}} = 2\sqrt{\lambda L} = \sqrt{4\lambda L}$.
What mass of \(\mathrm{Al}_2\mathrm{O}_3\) is produced from 18.5 g Al? \(4\mathrm{Al} + 3\mathrm{O}_2 \rightarrow 2\mathrm{Al}_2\mathrm{O}_3\)
Molar mass Al = 27 g/mol: moles Al = 18.5/27 = 0.685 mol. From stoichiometry: 4 mol Al β 2 mol AlβOβ, so mol AlβOβ = 0.685/2 = 0.3425 mol. Molar mass AlβOβ = 102 g/mol: mass = 0.3425Γ102 = 34.94 g β 34.9 g.
A business owner purchased a delivery vehicle and made these payments:
| Item | Amount ($) |
|---|---|
| Delivery vehicle purchase price | 28,000 |
| Fitting shelves inside the vehicle | 3,000 |
| Painting business logo on the vehicle | 2,000 |
What was the total capital expenditure?
Option C ($31,000) is correct.
Capital expenditure includes costs that bring a non-current asset into use or improve its capacity:
- Vehicle purchase price: $28,000
- Fitting shelves (improves functionality): $3,000
Total CapEx = $31,000
Painting the logo ($2,000) is revenue expenditure β a marketing cost that does not add to the asset's useful life.
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