A colony of bacteria grows exponentially The colony begins w

A colony of bacteria grows exponentially. The colony begins with 35 bacteria, but 5 hours after the beginning of the experiment, it has grown to 210 bacteria.

(a) Give a formula for the number of bacteria, y as a function of time t, where t is in hours.

Your answer begins with y = and contains the variable t. There should be no rounding in your final answer.

(b) How long does it take for the colony to triple in size?
Please report your answer to part (b) rounded to two decimal places.

Solution

y = y_o * e^k*t

where \"y\" is the population after time \"t\"

\"y_o\" is the initial population

\"e\" is a constant

\"t\" time taken

therefore

210 = 35 e^k*5

6 = e^5k                     // divided both sides by \"35\"

ln 6 = ln e^5k               // taking log on both sides

ln 6 = 5k

or k = ln6 / 5

a)

so the formula for the number of bacteria is

y = 35 * e^{((ln 6)/5 * t)}

y = 35 e^{((ln 6/5) * t)}

this will give the poplulation of bacteria at any time \"t\"

b)

so the population of bacteria has grown to 3 times

initial population = 35

final population = 3 times intial population = 3*35 = 105

105 = 35 e^kt

3 = e^{((ln 6)/5) * t}

taking log both sides

ln 3 = ln e^{((ln 6)/5) * t}

ln 3 = ln6/5 *t

5 ln 3 = ln 6 * t

t = 5 ln 3 / ln (6)

t = 5*1.098 / 1.791

t = 3.06 hrs