Concept explainers
Population Growth These exercises use the population growth model.
14. Bacteria Culture The count in a culture of bacteria was 400 after 2 hours and 25,600 after 6 hours.
- (a) What is the relative rate of growth of the bacteria population? Express your answer as a percentage.
- (b) What was the initial size of the culture?
- (c) Find a function that models the number of bacteria n(t) after t hours.
- (d) Find the number of bacteria after 4.5 hours.
- (e) After how many hours will the number of bacteria reach 50,000?
(a)
To find: The relative growth rate of the bacteria population if the bacteria population after
Answer to Problem 14E
The relative growth rate is
Explanation of Solution
Formula used:
Exponential growth model (relative growth rate) is,
Where,
Calculation:
Given that the bacteria population after
Substitute
Thus, the bacteria population at 2 hours is
The bacteria population after
Thus, the bacteria population at 6 hours is
Divide
Simplify the above expression,
Thus, the relative growth rate is
(b)
To find: The initial size of the culture of bacteria.
Answer to Problem 14E
The initial size of the bacteria population in the culture is
Explanation of Solution
Given that the bacteria population after
From part (a), the relative growth rate is
Substitute
Thus, the initial size of the bacteria population in the culture is
(c)
To find: The function for bacteria population.
Answer to Problem 14E
The function for bacteria population is
Explanation of Solution
From part (b), the initial population is
From part (a), the relative growth rate is
Substitute
Thus, the function for bacteria population is
(d)
To find: The number of bacteria after
Answer to Problem 14E
The bacteria population after
Explanation of Solution
From part (c), the population function is
Substitute
Thus, the bacteria population after
(e)
To find: The time in which the bacteria population reach to
Answer to Problem 14E
The bacteria population reach to
Explanation of Solution
From part (c), the population function is
Substitute
Simplify the above expression,
Thus, the bacteria population reach to
Chapter 4 Solutions
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