The number of bacteria in a culture is given by the function
[tex]$f(t)=970 e^{0.3 t}$[/tex]
where [tex]$t$[/tex] is measured in hours.
a) What is the exponential rate of growth of this bacterium population?
Your answer is $\square$ %
b) What is the initial population of the culture (at [tex]$t=0$[/tex])?
Your answer is $\square$
c) How many bacteria will the culture contain at time [tex]$t=2$[/tex]?
Your answer is $\square$
Answer (1)
The exponential rate of growth is found by multiplying the coefficient of t in the exponent by 100, resulting in 0.3 × 100 = 30% .
The initial population is determined by substituting t = 0 into the function, giving f ( 0 ) = 970 e 0.3 × 0 = 970 .
The population at t = 2 is calculated by substituting t = 2 into the function, yielding f ( 2 ) = 970 e 0.3 × 2 ≈ 1767 .
Therefore, the exponential rate of growth is 30% , the initial population is 970 , and the population at t = 2 is approximately 1767 .
Explanation
Understanding the Problem We are given the function f ( t ) = 970 e 0.3 t which models the number of bacteria in a culture, where t is measured in hours. We need to find the exponential rate of growth, the initial population, and the population at t = 2 .
Finding the Exponential Growth Rate The exponential rate of growth is the coefficient of t in the exponent, which is 0.3 . To express this as a percentage, we multiply by 100: 0.3 × 100 = 30 . Therefore, the exponential rate of growth is 30%.
Finding the Initial Population The initial population is the population at t = 0 . We substitute t = 0 into the function f ( t ) : f ( 0 ) = 970 e 0.3 × 0 = 970 e 0 = 970 × 1 = 970 . Therefore, the initial population is 970.
Finding the Population at t=2 To find the population at t = 2 , we substitute t = 2 into the function f ( t ) : f ( 2 ) = 970 e 0.3 × 2 = 970 e 0.6 . Since e 0.6 ≈ 1.8221 , we have f ( 2 ) ≈ 970 × 1.8221 ≈ 1767.437 . Since we are counting bacteria, we round to the nearest whole number, which is 1767.
Final Answer Therefore, the exponential rate of growth is 30%, the initial population is 970, and the population at t = 2 is approximately 1767.
Examples
Understanding exponential growth is crucial in various real-world scenarios. For instance, in epidemiology, it helps model the spread of infectious diseases, allowing healthcare professionals to predict infection rates and implement timely interventions. Similarly, in finance, it's used to calculate compound interest, enabling investors to estimate the growth of their investments over time. This concept also applies to environmental science, where it can model population growth or the decay of radioactive substances, aiding in conservation efforts and risk assessment.
