Question

The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in 7 hours, then in 35 hours its number would be ______.

Options

• 8 times the original

• 16 times the original

• 32 times the original

• 64 times the original

Answer 1

The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in 7 hours, then in 35 hours its number would be 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" = "k" "dt"`

Integrating on both sides, we get

log P = kt + c

When t = 0, P = P₀

∴ log P₀ = 0 + c ⇒ c = log P₀

∴ log P = kt + log P₀ 

`=> log "P"/"P"_0` = kt   ...(i)

When t = 7 hrs, P = 2P₀

∴ `log (2"P"_0)/"P"_0` = 7k

`=> "k" = (log 2)/7`

`∴ log "P"/"P"_0 = (log 2/7 "t")`    ...[From (i)]

When t = 35 hrs, we have

`log "P"/"P"_0 = (log 2)/7 xx 35 = 5 log 2 = log 32`

∴ P = 32P₀