MHT CET · Maths · Differential Equations

The micro-organisms double themselves in 3 hours. Assuming that the quantity increases at a rate proportional to it self, then the number of times it multiplies themselves in 18 years is

A 32
B 64
C 128
D 40

Verified Solution

Answer & Solution

Correct Answer

(B) 64

Step-by-step Solution

Detailed explanation

(B)
Let initial number of microorganisms be \(\mathrm{N}\). The microorganisms double themselves in 3 hours. \(\therefore\) Number of microorganisms after
3 hours \(=2 \times \mathrm{N}=2 \mathrm{~N}\)
o hours \(=2 \times 2 \mathrm{~N}=4 \mathrm{~N}\)
9 hours \(=2 \times 4 \mathrm{~N}=8 \mathrm{~N}\)
12 hours \(=2 \times 8 \mathrm{~N}=16 \mathrm{~N}\)
15 hours \(=2 \times 16 \mathrm{~N}=32 \mathrm{~N}\)
18 hours \(=2 \times 32 \mathrm{~N}=64 \mathrm{~N}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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