Math Problem Statement
A population of bacteria follows the continuous exponential growth model Upper P left parenthesis t right parenthesis equals Upper P 0 e Superscript kt, where t is in days. The relative (daily) growth rate is 3%. The current population is 888.
A) Find the growth model. (the function that represents the population after t days) P(t)= what?
B) Find the population exactly 3 weeks from now. Round to the nearest bacterium. The population in 3 weeks will be what?
C) Find the rate of change in the population exactly 3 weeks from now. Round to the nearest unit. The population will be increasing by about what bacteria per day exactly 3 weeks from now.
D) When will the population reach 6664? ROUND TO 2 DECIMAL PLACES. The population will reach 6664 about what days from now.
Solution
The given problem follows an exponential growth model described by:
where:
- is the initial population,
- is the growth rate (in decimal form), and
- is time (in days).
Given:
- Relative growth rate: (or 0.03 per day),
- Initial population .
Let's solve each part:
Part A) Find the growth model
The relative growth rate is , so the growth model becomes:
This is the function representing the population after days.
Part B) Find the population 3 weeks from now
Since 3 weeks is equivalent to days, substitute into the growth model:
We will calculate the exact population 3 weeks from now.
Part C) Find the rate of change in the population 3 weeks from now
The rate of change in the population can be found by taking the derivative of with respect to :
Substitute into this derivative to find the rate of change at that moment.
Part D) When will the population reach 6664?
We need to solve for when . Using the growth model:
Solve this equation for by isolating :
Now I'll calculate all the values for parts B, C, and D.Here are the results for each part of the problem:
Part A) Growth Model:
Part B) Population exactly 3 weeks from now:
The population in 3 weeks (21 days) will be approximately:
Part C) Rate of change in the population 3 weeks from now:
The population will be increasing at a rate of about:
Part D) When will the population reach 6664?
The population will reach 6664 bacteria in approximately:
Would you like any additional details or have further questions?
Here are 5 questions that expand on this information:
- How would the growth model change if the growth rate were doubled?
- What if the initial population were reduced to half? How would it affect the result in Part B?
- How can we calculate the doubling time for this population growth?
- What if the growth rate was continuously compounded hourly instead of daily?
- How can we calculate the population after 1 month (30 days) given this model?
Tip: Exponential growth models are useful for describing populations that grow proportionally to their size, such as bacteria or financial investments!
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Math Problem Analysis
Mathematical Concepts
Exponential Growth
Differential Calculus
Logarithmic Equations
Formulas
Exponential Growth Model: P(t) = P0 * e^(kt)
Rate of Change: dP/dt = P0 * k * e^(kt)
Logarithmic Equation: ln(A) = B
Theorems
Properties of Exponential Functions
Derivative of Exponential Functions
Logarithmic Properties
Suitable Grade Level
Grades 11-12