A growth medium is inoculated with 1,000 bacteria, which grow at a rate of $15 \%$ each day. What is the population of the culture 6 days after inoculation?
A. $y=1,000(1.15)^6 ; 2,313$ bacteria
B. $y=1,000(1.15)^7 ; 2,660$ bacteria
C. $y=1,000(1.5)^5 ; 7,594$ bacteria
D. $y=1,000(1.5)^6 ; 11,391$ bacteria
Answer (2)
The problem describes an exponential growth scenario. We calculate the population of bacteria after 6 days using the formula for exponential growth. The initial population is 1,000, the growth rate is 15%, and the time is 6 days. We plug these values into the formula and calculate the final population. Rounding to the nearest whole number, we find that the population is approximately 2 , 313 bacteria.
Explanation
Understanding the Problem We are given that a growth medium is inoculated with 1,000 bacteria, and the bacteria grow at a rate of 15% each day. We want to find the population of the culture 6 days after inoculation.
Stating the Formula The formula for exponential growth is given by: y = P ( 1 + r ) t where:
y is the final population
P is the initial population
r is the growth rate (as a decimal)
t is the time in days
Identifying the Values In this case, we have:
P = 1000
r = 15% = 0.15
t = 6
Calculating the Population Plugging these values into the formula, we get: y = 1000 ( 1 + 0.15 ) 6 y = 1000 ( 1.15 ) 6 y = 1000 ( 2.313060765625 ) y = 2313.060765625
Rounding to the Nearest Whole Number Since we are dealing with a population of bacteria, we should round to the nearest whole number. Therefore, the population of the culture 6 days after inoculation is approximately 2313 bacteria.
Examples
Exponential growth can be seen in various real-world scenarios, such as compound interest in finance. Imagine you invest $1,000 in an account that earns 15% interest annually. After 6 years, the investment will grow to approximately $2,313, demonstrating the power of exponential growth over time. This concept is also applicable in understanding population growth, spread of diseases, and other phenomena where quantities increase rapidly.
After inoculating a growth medium with 1,000 bacteria that grow at a rate of 15% per day, the population after 6 days is approximately 2,313. This is calculated using the exponential growth formula: y = 1000 ( 1.15 ) 6 . The correct answer is option A: y = 1 , 000 ( 1.15 ) 6 ; 2 , 313 bacteria.
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