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
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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