The number of bacteria in a petri dish doubles every 4 hours. If there are initially 200 bacteria.
a) How many bacteria will there be after 12 hours?
b) How many bacteria will there be after 2 days?
y = A bᵗ
The starting amount of bacteria is 100, so A = 100
To find b, plug in this equation, t = 4 = and y = 200 (since the population doubles in 4 hours)
200 = 100 ∙ b⁴
Divide both sides by 2
200 / 100 = b⁴
b⁴ = 2
b = ∜2
y = A bᵗ
y = 100 ∙ (∜2 )ᵗ
a)
After 12 h:
y = 100 ∙ (∜2 )¹²
Since (∜2 )¹² = 2³
100 ∙ 2³ = 100 ∙ 8 = 800
b)
2 days = 48 h
y = 100 ∙ (∜2 )⁴⁸
Since (∜2 )⁴⁸ = 2¹²
y = 100 ∙ 2¹² = 100 ∙ 4096 = 409 600
The previous post is wrong because I took the wrong starting value of the bacteria.
The solution should be written as follows:
y = A bᵗ
The starting amount of bacteria is 200, so A = 200
To find b, plug in this equation, t = 4 = and y = 400 (since the population doubles in 4 hours)
y = A bᵗ
y = 100 bᵗ
400 = 100 ∙ b⁴
Divide both sides by 4
400 / 100 = b⁴
b⁴ = 4
b = ∜4
b = √2
y = A bᵗ
y = 100 ∙ √2ᵗ
a)
After 12 h:
y = 100 ∙ ( √2 )¹²
Since ( √2 )¹² = 2⁶
100 ∙ 2⁶ = 100 ∙ 64 = 6 400
b)
2 days = 48 h
y = 100 ∙ ( √2 )⁴⁸
Since ( √2 )⁴⁸ = 2²⁴
y = 100 ∙ 2²⁴ = 100 ∙ 16 777 216 = 1 677 721 600
Ignore my first post.
To solve this problem, we can use the formula for exponential growth:
N(t) = N0 * 2^(t/d)
Where:
N(t) is the number of bacteria at time t
N0 is the initial number of bacteria
t is the elapsed time
d is the doubling time (time it takes for the number of bacteria to double)
a) How many bacteria will there be after 12 hours?
In this case, the doubling time is 4 hours, so we can plug the values into the formula:
N(12) = 200 * 2^(12/4)
Simplifying the exponent:
N(12) = 200 * 2^3
Calculating the exponential term:
N(12) = 200 * 8
N(12) = 1600
Therefore, there will be 1600 bacteria after 12 hours.
b) How many bacteria will there be after 2 days?
Since there are 24 hours in a day, there are 2 * 24 = 48 hours in 2 days.
Using the formula:
N(48) = 200 * 2^(48/4)
Simplifying the exponent:
N(48) = 200 * 2^12
Calculating the exponential term:
N(48) = 200 * 4096
N(48) = 819,200
Therefore, there will be 819,200 bacteria after 2 days.
To solve this problem, we can use the exponential growth formula:
N(t) = N(0) * (2^(t/d))
where:
N(t) is the number of bacteria at time t
N(0) is the initial number of bacteria
t is the time in hours
d is the doubling time in hours
a) To find the number of bacteria after 12 hours, we can substitute the given values into the formula:
N(t) = N(0) * (2^(t/d))
N(12) = 200 * (2^(12/4))
Now, we can calculate the result:
N(12) = 200 * (2^3)
N(12) = 200 * 8
N(12) = 1600
Therefore, there will be 1600 bacteria after 12 hours.
b) To find the number of bacteria after 2 days, we need to convert 2 days into hours, as the formula uses hours for time.
Since there are 24 hours in a day, we can calculate the number of hours in 2 days:
2 days * 24 hours/day = 48 hours
Now that we have the time (t), we can substitute it into the formula:
N(t) = N(0) * (2^(t/d))
N(48) = 200 * (2^(48/4))
Calculating the result:
N(48) = 200 * (2^12)
N(48) = 200 * 4096
N(48) = 819,200
Therefore, there will be 819,200 bacteria after 2 days.