A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 61 cells. Find the relative growth rate.

the relative growth rate k = ln( 8 )

Step-by-step explanation:

Given the data in the question;

Initial population = 61 cells

which divides into two cells every 20 minutes;

Now.

dy/dt = ky

where y(t) = y(0)

y(t) is the population of bacteria

t is time in hours

y(0) is initial population

we know that, Initial population y(0) = 61

time t = 20 min = ( 20 / 60 )hr = 1/3 hr

so, y(1/3) = 2( 61 ) = 122

Now,

y(t) = y(0)

y(1/3) = 61

122 = 61

divide both sides by 122

2 =

ln(2) = k × 1/3

k = 3ln(2)

k = ln( 2³ )

k = ln( 8 )

Therefore, the relative growth rate k = ln( 8 )

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step-by-step explanation:

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starting with:    

5 + x - 12 = 2x - 7   

5 + x - 12 - x = 2x - 7 - x (subtract x from both sides)   

5 - 12 = x - 7 (combine and cancel x's from both sides)   

5 - 12 + 7 = x - 7 + 7 (add 7 to both sides)   

0 = x (add numbers together on both sides)       

so in order for the equation to be true, x must be 0.   

trying x = -0.5   

5 + x - 12 = 2x - 7 (original equation)   

5 + -0.5 - 12 = 2(-0.5) - 7 (plug in value of x)   

5 + -0.5 - 12 = -1 - 7 (did multiplication)   

-7.5 = -8 (did addition on both sides)       

since -7.5 is obviously not equal to -8, the value of -0.5 for x is wrong.   

trying x = 0   

5 + x - 12 = 2x - 7 (original equation)   

5 + 0 - 12 = 2(0) - 7 (plug in value of x)   

5 + 0 - 12 = 0 - 7 (did multiplication)   

-7 = -7 (did addition on both sides)       

since -7 equals -7, that demonstrates that the value of 0 for x is correct.   

trying x = 1   

5 + x - 12 = 2x - 7 (original equation)   

5 + 1 - 12 = 2(1) - 7 (plug in value of x)   

5 + 1 - 12 = 2 - 7 (did multiplication)   

-6 = -5 (did addition on both sides)       

since -6 is obviously not equal to -5, the value of 1 for x is wrong.