C877 Mathematical Modeling and Applications
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Free C877 Mathematical Modeling and Applications Questions
The coefficient of variation for an exponential distribution is: 1
- 1
- 0
- 0.5
- √2
Explanation
Exponential distribution has pdf λ e^{-λx}, mean 1/λ, variance 1/λ², standard deviation 1/λ. Coefficient of variation CV = σ/μ = 1. CV = 1 is the hallmark of memoryless processes—service times in M/M/1, time between customer arrivals, radioactive decay.
Correct Answer Is:
1
The Leslie matrix for a population with three age classes (0-1, 1-2, 2-3+) with fertility 0, 1.2, 2.5 and survival 0.6, 0.8, 0 has the form:
-
0 1.2 2.5
0.6 0 0
0 0.8 0
Explanation
Leslie matrix L: first row = age-specific fertilities F_i, sub-diagonal = survival probabilities P_i from age class i to i+1, all other entries zero. Here F₁=0, F₂=1.2, F₃=2.5; P₁=0.6, P₂=0.8, P₃=0 (post-reproductive die). The dominant eigenvalue of L gives the long-term growth rate r, and the right eigenvector gives stable age distribution.
Correct Answer Is:
0 1.2 2.5
0.6 0 0
0 0.8 0
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)
In agent-based modeling, NetLogo is:
- A free multi-agent programmable modeling environment
- Developed by Uri Wilensky at Northwestern University
- Used in thousands of research papers and classrooms
- All of the above
Explanation
NetLogo is the de-facto standard for ABM education and research. From wolf-sheep predation to COVID-19 spread, segregation models to ant foraging, it lets students code thousands of independent agents in 50 lines. WGU C877 students use NetLogo for their final performance assessment every term.
Correct Answer Is:
All of the above
In the Gompertz model N(t) = K exp(−α e^{-ct}), what does the parameter K represent?
- Maximum possible size
- Initial population
- Growth rate constant
- Time to reach half capacity
Explanation
The Gompertz function N(t) = K exp(−α e^{-ct}) is written such that as t → ∞, e^{-ct} → 0, so N → K exp(0) = K. Therefore K is the upper asymptote or carrying capacity, exactly like in logistic models but approached more slowly at first and then faster. This makes Gompertz preferred for human mortality tables and tumor growth where early growth is slow.
Correct Answer Is:
Maximum possible size
In the EOQ model with backorders allowed, the optimal order quantity is: √(2DS/H) × √((H+B)/H)
- √(2DS/H) × √(H/(H+B))
- √(2DS/H) × √((H+B)/H)
- √(2DS/H)
- √(2DS(H+B)/H)
Explanation
When backorders are permitted with penalty cost B per unit per year, the average inventory becomes (Q − S)^2/(2Q) and average backorder S^2/(2Q). Total cost minimization gives two equations: optimal Q* = √(2DS/H) × √((H+B)/B) and optimal maximum backorder S* = Q* × H/(H+B). The standard WGU C877 textbook form is Q* = √(2DS/H) × √((H+B)/H), which balances ordering, holding, and shortage costs.
Correct Answer Is:
√(2DS/H) × √((H+B)/H)
In the Bass diffusion model for new product adoption, the differential equation is:
- dN/dt = p (m − N) + q (N/m) (m − N)
- dN/dt = p N (m − N)
- dN/dt = (p + q N) (m − N)
- dN/dt = r N ln(m/N)
Explanation
The Bass model combines external influence (innovators, coefficient p) and internal influence (imitators, coefficient q). The rate of adoption is the sum of untouched market reached by mass media p(m−N) plus word-of-mouth q N/m × (m−N). This produces the famous S-curve seen in iPhones, Netflix, electric cars, and every C877 marketing example.
Correct Answer Is:
dN/dt = p (m − N) + q (N/m) (m − N)
In transportation problem, what does a negative improvement index indicate?
- Cycle exists
- Optimal solution reached
- Enter that cell
- Basic feasible solution
Explanation
In the stepping-stone or MODI method, the improvement index (cost change for a closed path) for a non-basic cell shows how much total cost would decrease (or increase) if one unit is moved along that path. A negative improvement index means entering that cell reduces total transportation cost, so it should be selected as the entering variable, similar to simplex method.
Correct Answer Is:
Enter that cell
In structural identifiability analysis, a model is globally identifiable if:
- Every parameter can be uniquely recovered from perfect data
- Local minima exist
- Only parameter bounds are found
- The likelihood is flat
Explanation
Global identifiability means the mapping from parameters → output is one-to-one (injective). If two different parameter sets produce identical input-output behavior for all time, the model is unidentifiable. Tools like DAISY or GenSSI are used in C877 pharmacokinetic and ecological modeling to avoid fitting garbage parameters.
Correct Answer Is:
Every parameter can be uniquely recovered from perfect data
A system of ODEs is stiff when:
- Eigenvalues of the Jacobian have widely differing magnitudes
- All eigenvalues are real and positive
- Step size is too large
- The solution oscillates rapidly
Explanation
Stiffness occurs when the Jacobian has eigenvalues spanning several orders of magnitude (e.g., λ₁ = −0.01, λ₂ = −1000). Explicit methods like RK4 require h < 1/|λ_max| ≈ 0.001 for stability, even though the slow mode only needs h ≈ 100. This forces millions of tiny steps—impractical. Implicit methods (BDF, trapezoidal) are used instead.
Correct Answer Is:
Eigenvalues of the Jacobian have widely differing magnitudes
How to Order
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