Question

Help please!! A bacteria culture is started with 300 bacteria. After 4 hours, the population had grown to 500 bacteria. If the population grows exponentially. a: write a recursive formula for the number of bacteria. b: write an explicit formula for the number of bacteria. c: if this trend continues, how many bacteria will there be in 1 day?

Answer 1

a).
`P_(t)=(5)/(3)P_(t-1)`

b).
`A_(t)=300e^(0.1277* t)`

c). 6429 bacteria

Explanation:

Population of the bacteria grows exponentially.

Therefore, growth of the bacteria will be represented by the formula

`P_(t)=P_(0)e^(kt)`

Where
`P_(t)` = Population of the bacteria after time t

`P_(0)` = Initial population of the bacteria

k = growth constant

t = time taken for growth

Now we plug in the values in the formula

`P_(t)` = 500

`P_(0)` = 300

Time t = 4 hours

`500=300e^(4k)`

`e^(4k)=(500)/(300)`

Now we take the natural log (ln) on both the sides

`ln(e^(4k))=ln((500)/(300))`

4k(lne) = ln(500) - ln(300)

4k = 6.2146 - 5.7038

4k = 0.5108

k =
`k=(0.5108)/(4)=0.1277`

a). Recursive formula for the sequence formed by the bacterial growth

Since
`P_(t)=(1+r)P_(t-1)`

500 = (1 + r)300

`(1+r)=(5)/(3)`

r =
`(5)/(3)-1=(2)/(3)`

Therefore, the recursive formula will be

`P_(t)=(5)/(3)P_(t-1)`

b). Explicit formula for the number of the bacteria will be

`A_(t)=300e^(0.1277* t)`

c). We have to calculate the number of bacteria after 24 hours

`A_(t)=300e^(0.1277* 24)`

`A_(t)=300e^(3.0648)`

`A_(t)=300* 21.43`

`A_(t)=6429` bacteria