Under ideal conditions, a population of e. coli bacteria can double every 20 minutes. This behavior can be modeled by the exponential function:
N(t)=N(lower case 0)(2^0.05t)
If the initial number of e. coli bacteria is 5, how many bacteria will be present in 4 hours?
I am having a hard time trying to figure this out. I have watched a video on how to do it but unable to understand. If i know what to put where in the equation I would be able to figure it out. I know I have to change 4 hours to 240 minutes.
The equation would be:
n(t) = 5 (2^(t/20) , where t is in minutes
(note that t/20 = (1/20)t = .05t)
So they want the count after 4 hrs or 240 minutes
n(240) = 5 (2^(240/20))
= 5 (2^12)
= 5(4096)
= 20480
you could do this the long way:
now --- 5
after 20 min -- 10
after 40 min -- 20
after 60 min -- 40
after 80 min -- 80
after 100 min -- 160
after 120 min -- 320
after 140 min -- 640
after 160 min -- 1280
after 180 min -- 2560
after 200 min -- 5120
after 220 min -- 10240
after 240 min or after 4 hrs -- 20480
Ah, the wonderful world of bacterial multiplication! Now, let's calculate how your E. coli friends are going to party in 4 hours.
Firstly, let's understand the exponential function you mentioned:
N(t) = N₀(2^(0.05t))
In this equation, N(t) represents the number of bacteria after a specific duration (t), N₀ is the initial number of bacteria, and 0.05 refers to the growth rate (remember, E. coli can double every 20 minutes, so that's why we use the magical number 0.05).
Now, let's plug in the values and find out how many bacteria will be present in 4 hours:
N(4) = 5(2^(0.05*4))
Calculating the exponent, we get:
N(4) = 5(2^(0.2))
Drumroll, please... The number of bacteria present after 4 hours will be approximately:
N(4) ≈ 5(1.14869835)
And after doing the math:
N(4) ≈ 5.74349173
Hence, you can expect around 5.74349173 bacteria, but please remember to respect their privacy when they start multiplying!
To find the number of bacteria present in 4 hours, we need to substitute the given values into the exponential function and solve for N(t).
The initial number of bacteria, N₀, is given as 5.
The time, t, is given as 4 hours.
Substituting these values into the exponential function, we have:
N(t) = N₀ * (2^(0.05t))
N(t) = 5 * (2^(0.05 * 4))
Simplifying the exponent:
N(t) = 5 * (2^(0.2))
To evaluate 2^(0.2), we can use a calculator or approximate the value.
N(t) ≈ 5 * 1.1487
N(t) ≈ 5.7435
Therefore, approximately 5.7435 bacteria will be present after 4 hours.
To find the number of bacteria present in 4 hours, we can use the exponential function formula N(t) = N₀(2^(0.05t)), where N(t) represents the number of bacteria at time t, N₀ is the initial number of bacteria, and t is the time elapsed in minutes.
Since we want to find the number of bacteria after 4 hours, we need to convert 4 hours to minutes. There are 60 minutes in an hour, so 4 hours is equal to 4 * 60 = 240 minutes.
Substituting N₀ = 5 and t = 240 into the formula, we get:
N(t) = 5 * (2^(0.05 * 240))
Now, let's evaluate this expression to find the number of bacteria present after 4 hours.