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Q2:
If a population growing exponentially double in size in 3 years, what is the intrinsic rate of increase ((r)) of the population?
Solution :
Step 1: Theoretical Foundation of Exponential Growth
The exponential growth of a population is mathematically represented by the differential equation $\frac{dN}{dt} = rN$. Upon integration, this yields the equation for population size at a given time $t$:
$N_t = N_0 e^{rt}$
Where:
- $N_t$ = Population size at time $t$
- $N_0$ = Initial population size
- $e$ = Base of the natural logarithm ($\approx 2.71828$)
- $r$ = Intrinsic rate of natural increase
- $t$ = Time interval
Step 2: Assigning Given Variables
Based on the problem statement, we have the following parameters:
- Doubling time ($t$) = $3$ years
- Final population ($N_t$) = $2N_0$
Step 3: Algebraic Derivation for $r$
Substitute the known values into the growth equation:
$2N_0 = N_0 e^{r(3)}$
[By dividing both sides by $N_0$]:
$2 = e^{3r}$
[By taking the natural logarithm ($\ln$) on both sides to solve for the exponent]:
$\ln(2) = \ln(e^{3r})$
[Using the logarithmic identity $\ln(e^x) = x$]:
$\ln(2) = 3r$
$r = \frac{\ln(2)}{3}$
Step 4: Numerical Calculation
Given that $\ln(2) \approx 0.6931$:
$r = \frac{0.6931}{3}$
$r \approx 0.2310$
Visual Representation of Exponential Growth
Final Solution: The intrinsic rate of increase ($r$) for the population is approximately $0.231$ per year.
More Questions from Class 12 Biology Organisms and Populations EXERCISES
- Q1: List the attributes that populations possess but not individuals.
- Q1: List the attributes that populations possess but not individuals.
- Q2: If a population growing exponentially double in size in 3 years, what is the intrinsic rate of increase (\(r\)) of the population?
- Q3: Name important defence mechanisms in plants against herbivory.
- Q3: Name important defence mechanisms in plants against herbivory.
- Q4: An orchid plant is growing on the branch of mango tree. How do you describe this interaction between the orchid and the mango tree?
- Q4: An orchid plant is growing on the branch of mango tree. How do you describe this interaction between the orchid and the mango tree?
- Q5: What is the ecological principle behind the biological control method of managing with pest insects?
- Q5: What is the ecological principle behind the biological control method of managing with pest insects?
- Q6: Define population and community.
- Q6: Define population and community.
- Q7(a): Define the following terms and give one example for each: (a) Commensalism
- Q7(a): Define the following term and give one example for it: Commensalism
- Q7(b): Define the following terms and give one example for each: (b) Parasitism
- Q7(b): Define the following term and give one example for it: Parasitism
- Q7(c): Define the following terms and give one example for each: (c) Camouflage
- Q7(c): Define the following term and give one example for it: Camouflage
- Q7(d): Define the following terms and give one example for each: (d) Mutualism
- Q7(d): Define the following term and give one example for it: Mutualism
- Q7(e): Define the following terms and give one example for each: (e) Interspecific competition
- Q7(e): Define the following term and give one example for it: Interspecific competition
- Q8: With the help of suitable diagram describe the logistic population growth curve.
- Q8: With the help of suitable diagram describe the logistic population growth curve.
- Q9: Select the statement which explains best parasitism.
- Q9: Select the statement which explains best parasitism.
- Q10: List any three important characteristics of a population and explain.
- Q10: List any three important characteristics of a population and explain.
CBSE Solutions for Class 12 Biology Organisms and Populations
Chapters in CBSE - Class 12 Biology
- Reproduction in Organisms
- Human Reproduction
- Reproductive Health
- Principles of Inheritance and Variation
- Molecular Basis of Inheritance
- Evolution
- Human Health and Disease
- Strategies for Enhancement in Food Production
- Microbes in Human Welfare
- Biotechnology: Principles and Processes
- Biotechnology and Its Applications
- Ecosystem
- Biodiversity and Conservation
- Environmental Issues
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