A certain culture initially contains 10,000 bacteria and increase by 20% after every hour.
A) What will be the formula for numbers N(t) of bacteria after "t" hours?
B) How many bacteria are in culture at the end of 10 hours?
Every hour, the amount is increased by 20%, meaning it's multiplied by 1.2
So,
N(t) = N(initial)*((1.2)^t)
= 10000*((1.2)^t)
So every hour the number gets multiplied with an additional 1.2
When t = 10,
N(10) = 10000*((1.2)^10)
= 10000*6.191
= 61,910
Exponential Growth:
N(t) = N ∙ ( 1 + r )ᵗ
N = initial amount
r = growth rate
t = time elapsed
In this case:
N = 10,000
r = 20% = 20 / 100 = 0.2
A)
N(t) = 10,000 ∙ ( 1 + 0.2 )ᵗ
N(t) = 10,000 ∙ 1.2 ᵗ
B)
N(10) = 10,000 ∙ 1.2 ¹⁰ = 10,000 ∙ 1.2 ¹⁰ = 10,000 ∙ 6.1917364224 = 61,917.364224
Approx.
62,000
A) The formula for the number of bacteria N(t) after "t" hours can be calculated using exponential growth formula:
N(t) = N0 * (1 + r)^t
Where:
N(t) is the number of bacteria after t hours
N0 is the initial number of bacteria (10,000 in this case)
r is the growth rate per hour (20% or 0.2 as a decimal)
t is the number of hours
Therefore, the formula for N(t) becomes:
N(t) = 10,000 * (1 + 0.2)^t
B) Now, we need to find out the number of bacteria in the culture at the end of 10 hours.
Using the formula from part A:
N(10) = 10,000 * (1 + 0.2)^10
Calculating further:
N(10) = 10,000 * (1.2)^10
N(10) = 10,000 * 6.1917364224
N(10) ≈ 61,917.36
Therefore, there are approximately 61,917 bacteria in the culture at the end of 10 hours.
A) To find the formula for the number of bacteria after "t" hours, we need to consider that the number of bacteria increases by 20% after every hour.
Let's start with the initial number of bacteria, which is 10,000. After 1 hour, the number of bacteria increases by 20%, which means it becomes 10,000 + 20% of 10,000.
20% of 10,000 is calculated by multiplying 10,000 by 0.20, which gives us 2,000. So, after 1 hour, the total number of bacteria becomes 10,000 + 2,000 = 12,000.
Now, after 2 hours, the number of bacteria increases by 20% again, which means it will become 12,000 + 20% of 12,000.
We can continue this pattern for any number of hours. The formula to calculate the number of bacteria after "t" hours is:
N(t) = 10,000 * (1 + 0.20)^t
Here, (1 + 0.20) represents an increase of 20% after each hour, and "t" denotes the number of hours.
B) Now, let's use the formula to find the number of bacteria at the end of 10 hours. Plug in "t = 10" into the formula:
N(10) = 10,000 * (1 + 0.20)^10
= 10,000 * (1.20)^10
To calculate (1.20)^10, you can raise 1.20 to the power of 10, either by using a calculator or by multiplying 1.20 by itself 10 times. Doing the calculation gives us approximately 61,917.98.
Now, multiply 10,000 by 61,917.98 to get the final answer:
N(10) ≈ 10,000 * 61,917.98
≈ 619,179,800
Therefore, there will be approximately 619,179,800 bacteria in the culture at the end of 10 hours.