Question

The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in 5 hours then in 25 hours, its number would be

Options

• 8 times the original

• 16 times the original

• 32 times the original

• 64 times the original

Answer 1

32 times the original

Explanation:

Let P₀ be the initial population and let the population after t years be P.

Then `(dP)/(dt)` = kP ⇒ `(dP)/P` = kdt

On integrating, we have log P = kt + c

At t = 0, P = P₀

Therefore, log P₀ = 0 + c.

So, log P = kt + log P₀ ⇒ `log  P/P_0` = kt

When t = 5 h, P = 2P₀.

Therefore, `log  (2P_0)/P_0` = 5 k ⇒ k = `log  2/5`

Therefore, `log  P/P_0 = log  2/5  t`

When t = 25 h, we have

`log  P/P_0 = log  2/5 xx 25` = 5 log 2 = log 32

Hence, P = 32 P₀.