Question

A bacteria culture starts with 460 bacteria and grows at a rate proportional to its size. After 2 hours there will be 920 bacteria. (a) Express the population after t hours as a function of t.

Answer 1

Answer: Check explanation.

Step-by-step explanation:

Important things to note from the question: the total number the bacteria stated with is =460. After time(t), which is two hours passed, the numbers of bacteria increased to= 920.

Now, let us delve into the solution.

The number of bacteria at any time,t= N(t). And N(t) = e^Xt + V. Where V = ln 460 and X can be calculated below;

total number of bacteria remaining after the two hours, 920 = e^2X. e^460.

So, we make Exponential 2X the subject of the formula;

Therefore, e^2X = 920/460.

====> e^2X = 2.

The exponential goes into the right side, turns into 'ln'.

====> 2X = ln 2.

Therefore, X= 1/2 ×ln 2.

OR

X= 0.5 × 0.69315 = 0.35.

Therefore, Expressing the population after t hours as a function of t;

N(t) = 460 × (e^1/2 ln 2)t.

OR

N(t)= 460× e^0.35 t.

CONCLUSION: With this we can slot in any value of time,t to determine the value of the number of bacteria at any time,t(N)