Question

The rate of growth of bacteria is proportional to the number present. If initially, there are 1000 bacteria and the number doubles in 1 hour, the number of bacteria after `21/2`  hours will be ______. `(sqrt(2) = 1.414)`

Options

• 5464

• 5636

• 5656

• 6565

Answer 1

The rate of growth of bacteria is proportional to the number present. If initially, there are 1000 bacteria and the number doubles in 1 hour, the number of bacteria after `21/2`  hours will be 5656. `(sqrt(2) = 1.414)` 

Explanation:

Let 'x' be the number of bacteria present at time 't'.

∴ `("d"x)/("d"y) oo x`

∴ `("d"x)/"dt"` = kx

Integrating on both sides, we get

log x = kt+ c

When t = 0, x = 1000

∴ log (1000) = k(O) + c

⇒ c = log (1000)

∴ log x = kt + log (1000)  .......(i)

When t = 1, x = 2000

∴ log (2000) = k(1) + log (1000)

⇒ k = `log(2000/1000)` = log 2

∴ log x = t log 2 + log (1000)  ......[From(i)]

When t = `2 1/2 = 5/2`, we have

log x = `(5/2) log 2 + log (1000)`

= `log (2^(5/2)) + log (1000)`

= `log (4sqrt(2)) + log (1000)`

= `log (4000sqrt(2))`

= log (4000 x 1.414)

∴ log x = log (5656)

⇒ x = 5656