A population of bacteria begins with 15 strands and doubles every 4 hours. This can be modeled as [tex]$15(2)^{\frac{t}{4}}$[/tex], where t is time in hours. How many strands of bacteria are present at 12 hours?
A. 120 strands of bacteria
B. 400 strands of bacteria
C. 252 strands of bacteria
D. 200 strands of bacteria
Answer (1)
Substitute t = 12 into the equation 15 ( 2 ) 4 t .
Simplify the exponent: 4 12 = 3 .
Calculate 2 3 = 8 .
Multiply 15 by 8 to get the final answer: 15 × 8 = 120 . The number of bacteria strands present at 12 hours is 120 .
Explanation
Understanding the Problem We are given a formula that models the growth of a bacteria population: 15 ( 2 ) 4 t , where t is the time in hours. We want to find the number of bacteria strands present at t = 12 hours.
Substituting the Value of t To find the number of bacteria strands at 12 hours, we substitute t = 12 into the formula: 15 ( 2 ) 4 12 .
Simplifying the Exponent First, we simplify the exponent: 4 12 = 3 .
Calculating the Power Now we calculate 2 3 : 2 3 = 2 × 2 × 2 = 8 .
Multiplying to Find the Final Answer Finally, we multiply 15 by 8: 15 × 8 = 120 .
Stating the Answer Therefore, the number of bacteria strands present at 12 hours is 120.
Examples
Understanding exponential growth, as demonstrated in this bacteria problem, is crucial in various real-world scenarios. For instance, it helps in predicting population increases, calculating compound interest on investments, and modeling the spread of diseases. By grasping how quantities double over time, one can make informed decisions in finance, public health, and environmental planning, ensuring better preparedness and strategic resource allocation.
