C877 Mathematical Modeling and Applications
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Free C877 Mathematical Modeling and Applications Questions
In the M/M/1 queueing model, the average number of customers in the system L is ρ / (1 − ρ)
- ρ / (1 − ρ)
- λ / (μ − λ)
- ρ² / (1 − ρ)
- λ / μ
Explanation
M/M/1 assumes Poisson arrivals rate λ, exponential service rate μ, single server, infinite queue. Traffic intensity ρ = λ/μ < 1. By Little’s law L = λ W, and steady-state waiting time W = 1/(μ − λ), so L = λ/(μ − λ) = ρ/(1 − ρ). This is the classic result every operations research student memorizes—used for bank counters, call centers, and cloud servers.
Correct Answer Is:
ρ / (1 − ρ)
What is the carrying capacity in the logistic growth model dP/dt = rP(1 - P/K)?
- 200
- 9
- 1
- 0.2
Explanation
The logistic growth equation dP/dt = rP(1 - P/K) models population growth where the rate slows as the population P approaches the environmental carrying capacity K. In the standard form, K is the maximum sustainable population, appearing as the denominator in the limiting term (1 - P/K). For the given equation P(t) = 200 / (1 + 9e^{-0.2t}), rewriting it in logistic form reveals K = 200, as the population asymptotically approaches 200 over time. This represents the upper limit set by resources, a key concept in bounded growth models versus unlimited exponential growth.
Correct Answer Is:
200
In the Verhulst logistic model with harvesting at constant rate H, the DE becomes dP/dt = rP(1 − P/K) − H. The maximum sustainable yield (MSY) occurs at population level: K/2
- K/2
- K/4
- K
- 0
Explanation
The yield Y(H) = H (equilibrium harvested population). Equilibrium: P = K(1 − H/(rK)) → Y(H) = H K (1 − H/(rK)). This is a downward parabola in H with maximum at H_MS = rK/4, achieved when P = K/2. Harvesting at exactly rK/4 gives the highest long-term sustainable catch—classic fisheries management result (Schaefer model). Harvest more and the population collapses.
Correct Answer Is:
K/2
The Gauss-Seidel method differs from Jacobi because:
- It uses newly computed values immediately
- It uses over-relaxation
- It is explicit
- It only works for tridiagonal matrices
Explanation
Gauss-Seidel updates each point using the most recent values available (left and upper neighbors already updated in the same sweep), roughly halving the number of iterations needed compared to Jacobi. Successive over-relaxation (SOR) takes this further with a relaxation parameter ω > 1.
Correct Answer Is:
It uses newly computed values immediately
In the SI epidemic model without vital dynamics, dI/dt = β I (N − I)/N, the fraction eventually infected is:
- 1 − e^{−R₀ S₀/N}
- 1 − e^{−R₀}
- 1 − S₀/N
- R₀
Explanation
The SI model has no recovery or death, so infection spreads until no susceptibles remain. Separating variables gives ∫ dI/(I(N−I)) = (β/N) ∫ dt, which integrates to (1/N) ln(I/(N−I)) = (β/N) t + C. Using initial I(0) = I₀, the transcendental equation at t→∞ is S∞/N = e^{−R₀ I∞/N}. Since I∞ + S∞ = N, we get infected fraction = 1 − e^{−R₀ (1 − S∞/N)}, but the standard final-size relation is 1 − S∞/N = 1 − e^{−R₀ (1 − S∞/N)}, usually solved numerically. The closed-form approximation for large R₀ is 1 − e^{−R₀}, but exact is 1 − e^{−R₀ S₀/N} only when S₀ ≈ N
Correct Answer Is:
1 − e^{−R₀}
A population grows according to the differential equation dN/dt = 0.05N(1 - N/500). What is the intrinsic growth rate r?
- 0.05
- 500
- 0.05/500
- 1/500
Explanation
The standard logistic differential equation is dN/dt = rN(1 - N/K), where r is the intrinsic growth rate (maximum per capita growth when population is small) and K is the carrying capacity. Comparing to dN/dt = 0.05N(1 - N/500) shows the coefficient of N is r = 0.05, while K = 500. The intrinsic rate r determines how quickly the population grows near zero, independent of density-dependent limitations, and is crucial for comparing growth potential across species or models.
Correct Answer Is:
0.05
The 1D elementary cellular automaton Rule 90 produces:
- The Sierpinski triangle pattern
- Random noise
- Solid blocks
- Left-moving triangles
Explanation
Rule 90 XORs left and right neighbors: new cell = left XOR right. Starting from a single 1, it generates the Sierpinski cellular automaton—self-similar triangles with fractal dimension log₂(3) ≈ 1.58. This is the classic example of complex emergent patterns from dead simple rules.
Correct Answer Is:
The Sierpinski triangle pattern
A bacteria population follows exponential decay with half-life of 4 hours. What is the decay constant k in dN/dt = -kN?
- 0.1733 per hour
- 0.25 per hour
- 0.6931 per hour
- 4 per hour
Explanation
For exponential decay N(t) = N_0 e^{-kt}, the half-life T_{1/2} satisfies N(T_{1/2}) = N_0 / 2, so 1/2 = e^{-k T_{1/2}}, leading to k = ln(2) / T_{1/2} ≈ 0.6931 / T_{1/2}. With T_{1/2} = 4 hours, k = ln(2)/4 ≈ 0.1733 per hour. This constant k represents the proportional rate of decline, fundamental in modeling radioactive decay, drug elimination, or cooling processes where the rate is proportional to the current amount.
Correct Answer Is:
0.1733 per hour
In goal programming, if we have multiple goals with priorities P1 > P2 > P3, the method used is:
- Preemptive goal programming
- Non-preemptive (weighted) goal programming
- Lexicographic ordering
- Archimedean goal programming
Explanation
When goals have strict priority levels (P1 must be satisfied before even considering P2), we use preemptive (lexicographic) goal programming. The algorithm minimizes deviation from the highest priority first, then fixes that value and moves to the next priority level hierarchical objectives like “first minimize fatalities, then minimize cost”.
Correct Answer Is:
Preemptive goal programming
A predator-prey system is modeled by: dx/dt = 2x − 0.01 xy dy/dt = −0.5y + 0.0002 xy What is the meaning of the term −0.01 xy in the prey equation?
- Predator death rate
- Prey growth in absence of predator
- Predation rate
- Predator growth efficiency
Explanation
In Lotka-Volterra, prey equation is dx/dt = ax − bxy, where ax is intrinsic growth and −bxy is loss due to predation (proportional to encounters). Here a = 2, b = 0.01, so −0.01 xy represents the rate at which prey are eaten per predator per prey, i.e., the predation rate coefficient.
Correct Answer Is:
Predation rate
How to Order
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