Question

NASA scientists discovered a planet in December of 2004. After observation, they have come to a conclusion that the population doubles in January of every third year and due to calamities the population decreases to one-sixth in March of every 6th year. If the population in December of 2022 is 512 , then what was the population when the planet was discovered?
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Answer 1

Step-by-step explanation:

so, there is a population in that planet. right ?

that means they discovered a populated planet.

anyway,

18 years

p0 is the start population.

p0 × 2 after 3 years

p0 × 2 × 2 = p0 × 4 after 6 years

p0 × 4 / 6 after 6 years

(p0 × 4 / 6) × 2 after 9 years

(p0 × 4 / 6) × 4 after 12 years = p0 × 8/3 after 12 years

(p0 × 8/3) / 6 after 12 years = p0 × 8/18 = p0 × 4/9

p0 × 4/9 × 2 after 15 years

p0 × 4/9 × 4 after 18 years

p0 × 4/9 × 4 / 6 after 18 years = p0×16/54 = p0 × 8/27 = 512

p0 = 512×27/8 = 1728

p(n) = p0 × (2/3)^(n/6)

with n being years.

because

the population changes by a factor 2 every 3 years = 2^(n/3)

and by a factor 1/6 every 6 years = (1/6)^(n/6).

so,

p(n) = p0 × 2^(n/3) × (1/6)^(n/6)

2^(n/3) = 2^(2n/6) = 2²^(n/6) = 4^(n/6)

therefore,

p(n) = p0 × 4^(n/6) × (1/6)^(n/6) =

= p0 × (4/6)^(n/6) = p0 × (2/3)^(n/6)