Business Calculus Quiz: Interpreting De Solutions

The total revenue $R(t)$ in millions of dollars from a new streaming service is modeled by the Gompertz function $R(t) = 500e^{-2e^{-0.3t}}$, where $t$ is the number of years since launch. What is the projected maximum total revenue for this service according to the model?

A retirement fund's value $A(t)$ is modeled by $dA/dt = 0.06A - W$, where $A$ is the fund value in dollars, $t$ is time in years, the interest rate is 6% compounded continuously, and $W$ is a constant annual withdrawal amount. What is the interpretation of the withdrawal amount $W$ that makes $dA/dt = 0$?

The market share $P(t)$ for a new product is modeled by $P(t) = \frac{M}{1 + Ae^{-Mkt}}$, where $M$ is the total market size and $k$ is a growth constant. The parameter $A$ is determined by the initial market share, $P_0$.

Which statement correctly describes the business implication of a larger value of the parameter $A$?

A new software's market share, $S(t)$, as a proportion of the total market, is modeled by the differential equation $dS/dt = 0.4S(1-S)$, where $t$ is time in years. At what market share level is the company's software gaining market share most rapidly?

The balance $B(t)$ of a business loan after $t$ years is modeled by the solution $B(t) = (B_0 - P/r)e^{rt} + P/r$, where $B_0$ is the initial loan amount, $r$ is the annual interest rate compounded continuously, and $P$ is the annual payment amount made continuously.

The specific solution for a company's loan is given by $B(t) = 250000e^{0.05t} + 500000$. Based on this model, what was the company's annual payment amount?

The concentration $C(t)$ of a chemical in a manufacturing vat is modeled by the differential equation $dC/dt = 20 - 0.4C$, where $C$ is in grams per liter and $t$ is in hours. The process starts with a vat of pure solvent ($C(0)=0$). Which statement best describes the role of the term $-0.4C$?

A startup's user base $N(t)$ grows according to $\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) - \alpha N$ where $r = 0.4$, $K = 10000$, and $\alpha = 0.1$ represents user churn rate. The equilibrium solutions satisfy $N^* = 0$ or $N^* = K\left(1 - \frac{\alpha}{r}\right)$. What is the business interpretation of the condition $\alpha > r$?

A technology adoption model uses $\frac{dA}{dt} = k\sqrt{A}(M - A)$ where $A(t)$ is the number of adopters and $M = 10000$ is market size. The solution is $A(t) = M\left(\frac{C e^{k\sqrt{M}t} - 1}{C e^{k\sqrt{M}t} + 1}\right)^2$ where $C$ depends on initial conditions. If $A(0) = 100$ and $k = 0.002$, what does the term $k\sqrt{M}$ represent in the adoption dynamics?

A manufacturing company's production efficiency $E(t)$ follows $\frac{dE}{dt} = \alpha E(1-E) - \beta E^2$ where $\alpha = 0.6$ and $\beta = 0.2$. The equilibrium solutions are $E^* = 0$ and $E^* = \frac{\alpha - \beta}{\alpha} = \frac{2}{3}$. If the company starts with $E(0) = 0.8$ (80% efficiency), what does the model predict about long-term efficiency?

The number of people $N(t)$ who have adopted a new fashion trend is modeled by a logistic differential equation. The solution curve $N(t)$ has an inflection point at time $t=T$. What is the business significance of the time $T$?

The weekly sales $S(t)$ of a product, in thousands of units, $t$ weeks after a promotional event are modeled by $S(t) = 50te^{-kt}$, where the parameter $k > 0$ represents the decay effect of the promotion. A marketing analyst suggests that a more memorable campaign would have a smaller value of $k$. How would a smaller positive value of $k$ affect the timing of peak sales?

A new employee's productivity $P(t)$, measured in units assembled per hour, is modeled by the solution $P(t) = 80 - 50e^{-0.25t}$ after $t$ weeks of training. What was the employee's initial rate of increase in productivity?

The proportion $P(t)$ of employees in a large corporation who have heard a rumor after $t$ days is given by the logistic function $P(t) = \frac{1}{1 + Ae^{-kt}}$. The parameter $k$ is a positive constant. How does the value of the parameter $k$ relate to how the rumor spreads?

The weekly sales rate $S(t)$ (in thousands of units) for a product $t$ weeks after an advertising campaign ends is modeled by the solution $S(t) = 15 + 85e^{-0.2t}$. Which of the following statements best interprets the long-term sales behavior predicted by this model?

The value $V(t)$ of a piece of manufacturing equipment, in thousands of dollars, is modeled by the differential equation $dV/dt = -0.12V$. Which statement provides the correct business interpretation of this model?

An inventory model follows $\frac{dI}{dt} = -aI - b$ where $I(t)$ is inventory level, $a = 0.05$ (spoilage rate), and $b = 200$ (constant demand rate). The solution is $I(t) = \left(I_0 + \frac{b}{a}\right)e^{-at} - \frac{b}{a}$. If the company restocks to 6000 units whenever inventory hits 1000 units, and no additional deliveries occur between restocking, what happens to the time between restocking cycles as this process continues?

A viral marketing campaign follows $\frac{dN}{dt} = \beta N(P-N) - \gamma N$ where $N(t)$ is the number of people who have heard about the product, $P = 50000$ is the target population, $\beta = 10^{-6}$, and $\gamma = 0.02$ represents the rate people forget. The campaign succeeds if $N$ can reach at least 60% of $P$. What is the minimum condition for campaign success?

A pharmaceutical company models drug concentration in blood using $\frac{dC}{dt} = -kC + R$ where $C(t)$ is concentration (mg/L), $k = 0.3 \text{ hr}^{-1}$, and $R = 12 \text{ mg/L/hr}$ is the infusion rate. The solution is $C(t) = \frac{R}{k} + \left(C_0 - \frac{R}{k}\right)e^{-kt}$. If steady-state concentration must not exceed 45 mg/L for safety, and the patient starts with zero drug concentration, after how many hours will the concentration first reach 95% of steady-state?