Applications of Differential Equations
Description: This quiz covers various applications of differential equations, including population growth, radioactive decay, and more. | |
Number of Questions: 5 | |
Created by: Aliensbrain Bot | |
Tags: differential equations applications population growth radioactive decay |
Consider a population of rabbits that grows at a rate proportional to its size. If the initial population is 100 rabbits and the population doubles in 10 years, what is the population after 20 years?
A radioactive substance decays at a rate proportional to the amount present. If the half-life of the substance is 10 years, what percentage of the original amount remains after 20 years?
A spring-mass system is described by the differential equation $m\frac{d^2x}{dt^2} + kx = 0$, where $m$ is the mass, $k$ is the spring constant, and $x$ is the displacement from the equilibrium position. If the mass is 1 kg, the spring constant is 10 N/m, and the initial displacement is 1 meter, what is the amplitude of the resulting motion?
A tank initially contains 100 gallons of pure water. A salt solution with a concentration of 0.5 pounds per gallon is pumped into the tank at a rate of 10 gallons per minute, and the well-mixed solution is pumped out at the same rate. What is the amount of salt in the tank after 10 minutes?
A predator-prey model is given by the system of differential equations $\frac{dx}{dt} = x(1 - \frac{x}{K}) - \alpha xy$ and $\frac{dy}{dt} = -y(1 - \frac{y}{N}) + \beta xy$, where $x$ is the population of prey, $y$ is the population of predators, $K$ is the carrying capacity of the environment for the prey, $N$ is the carrying capacity of the environment for the predators, $\alpha$ is the attack rate of the predators, and $\beta$ is the conversion efficiency of the prey into predators. If $K = 1000$, $N = 500$, $\alpha = 0.01$, and $\beta = 0.005$, what is the equilibrium point of the system?