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- But our expression is $ \frac{1}{x} + \frac{4}{y} + \frac{9}{z} = \frac{1^2}{x} + \frac{2^2}{y} + \frac{3^2}{z} $
- Apply Titu’s Lemma:
- \frac{1^2}{x} + \frac
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Now divide by km/light-year: 2.083 × 10¹⁹ / 9.46 × 10¹² ≈ 2.203 × 10⁶ light-years
#### 2203000
An epidemiologist models a virus spreading in a population of 2 million. The infection rate doubles every 3 days, starting with 10 cases. After how many days will at least half the population (1 million) be infected?
Use exponential growth: cases = 10 × 2^(t/3)
Set 10 × 2^(t/3) ≥ 1,000,000
Divide both sides by 10: 2^(t/3) ≥ 100,000
Take log base 2: t/3 ≥ log₂(100,000) ≈ log₂(10⁵) = 5 × log₂(10) ≈ 5 × 3.3219 = 16.6095
So t ≥ 3 × 16.6095 ≈ 49.8285
Thus, after 50 days, cases exceed 1 million.
An industrial designer creates a circular sustainable packaging prototype with inner radius 8 cm and outer radius 10 cm. If the material thickness is uniform and the total volume of material used is 144π cm³, what is the thickness of the packaging?