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Bacteria grow in a culture at a rate proportional to the amount present. Initially, 300 strands of the bacteria are in the culture and after two hours that number has grown by 20 percent. Find (a) an expression for the approximate number of strands in the culture at any time t and (b) the time needed for the bacteria to double its initial size.

Respuesta :

princessesther2011

Answer:

(a) number of strands (n) = time (t) ÷ proportionality constant (k)

(b) The time needed for the bacterial to double its initial size is 3.36 hours.

Explanation:

(a) Let the rate (time) be represented by t and the amount (number) of strands of bacteria be represented by n

t is proportional to n, therefore, t = kn (k is the proportionality constant)

Since t = kn, then, n = t/k

(b) Initial amount of strands = 300

Amount of strands after 2 hours = 300 + (300 × 20/100) = 300 + 60 = 360

k = t/n = 2/360 = 0.0056 hour/strand

Double of the initial size is 600 (300×2 = 600)

Time (t) needed for the bacterial to double its initial size = kn = 0.0056×600 = 3.36 hours

raphealnwobi

It would take 7.6 hours for the bacteria to double its initial size.

An exponential function is given by:

y = abˣ

where a is the initial value of y and b is the multiplier.

Let y represent the number of strands after time t.

There are initially 300 strands of bacteria, hence: a = 300

Since the bacteria has grown by 20 percent after 2 hours, hence:

b = 100% + 20% = 1.2

a) The expression is given by:

[tex]y = 300(1.2)^{\frac{1}{2} t[/tex]

b) To double it size, y = 600, hence:

[tex]600=300(1.2)^\frac{t}{2} \\\\2=(1.2)^\frac{t}{2} \\\\\frac{t}{2} ln(1.2)=ln(2)\\\\\frac{t}{2} =3.8\\\\t=7.6\ hours[/tex]

Hence it would take 7.6 hours for the bacteria to double its initial size.

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