詮釋微分方程的解：练习题

The population of a species is modeled by a differential equation. The solution is found to be \( P = \frac{1200}{1 + 3e^{-0.5t}} \), where \( P \) is the population and \( t \) is the time in years.
Interpreting the solution of a differential equation, what is the carrying capacity (the limit of the population as \( t \to \infty \)) of this environment?

A quantity \( N \) satisfies the differential equation \( \frac{dN}{dt} = -k(N - 50) \), where \( k \) is a positive constant. Initially, \( N = 200 \). When interpreting the solution of this differential equation, what is the behavior of \( N \) as \( t \to \infty \)?

A scientist is interpreting the solution of a differential equation used to model the temperature \( T \) (in \( ^\circ \text{C} \)) of a cooling metal rod over time \( t \) (in minutes). The solution is given by \( T = 25 + 75e^{-0.1t} \).

Which of the following statements is not a valid interpretation of this model?

The velocity \( v \) of a parachutist is modeled by the solution \( v = 20 - 20e^{-t} \). (i) Describe the motion of the parachutist as \( t \to \text{∞} \). (ii) Identify a limitation of this model if the parachutist were to deploy a parachute midway through the fall.

A physicist models the cooling of a component using the differential equation \( \frac{d\theta}{dt} = -k(\theta - 20) \), where \( \theta \) is the temperature in \( ^\circ \text{C} \) and \( t \) is time in minutes.
The solution to this equation is \( \theta = 20 + Ae^{-kt} \).
(i) State the physical meaning of the constant 20 in this model.
(ii) Identify one limitation of using this model to predict the temperature over a very long period if the surrounding temperature fluctuates.

The rate of change of the volume of water, \(V\) m\(^3\), in a leaking tank at time \(t\) hours is modeled by the differential equation \(\frac{dV}{dt} = -0.1(V - 2)\). The tank initially contains 10 m\(^3\) of water.
(a) Show by integration that the solution to this differential equation is \(V = 2 + 8e^{-0.1t}\).
(b) Interpreting the solution of this differential equation:
(i) Find the volume of water in the tank after 5 hours, giving your answer to 2 decimal places.
(ii) Describe the long-term behavior of the volume of water in the tank as \(t \to \text{∞}\).
(iii) State one limitation of this model in a real-world context if the tank were to be refilled at a constant rate.

The velocity, \(v\) ms\(^{-1}\), of a skydiver at time \(t\) seconds after jumping from a plane is modelled by the differential equation:
\(\frac{dv}{dt} = 10 - 0.2v\)

(a) Given that the skydiver's initial velocity is 0 ms\(^{-1}\), solve the differential equation to show that \(v = 50(1 - e^{-0.2t})\).

(b) Describe the motion of the skydiver as \(t \to \infty\), and state the terminal velocity predicted by this model.

(c) Calculate the time taken for the skydiver to reach 90% of their terminal velocity, giving your answer to 3 significant figures.

(d) In a real-world scenario, the air resistance might be proportional to \(v^2\) rather than \(v\). Explain how the solution of the differential equation would differ if the model was changed to \(\frac{dv}{dt} = 10 - kv^2\), where \(k\) is a positive constant.