DP Mathematics: Applications and Interpretation Questionbank

Find the population of rabbits $1$ year after they were introduced.

Alessia estimates that the mackerel population density increases by a factor of three every two years. Show that $\alpha =0.549$ to three significant figures.

Show that the expression for the mackerel population density after $t$ years is $M=M_0{\text{e}}^{\alpha t}$

Write down the differential equation for $M$ that models this situation.

Show that $V={\left(20-\frac{t}{5}\right)}^2$.

Find the time taken for the tank to empty.

Find the equation of the regression line of $h$ on $t$.

Suggest why Eva’s use of the linear regression equation in this way could be unreliable.

Find the equation of the least squares quadratic regression curve.

Find the value of $k$.

Hence, write down a suitable domain for Eva’s function $h\left(t\right)=p{t}^2+qt+r$.

Interpret the meaning of parameter $a$ in the context of the model.

By solving the differential equation $\frac{dh}{dt}=-R^2\sqrt{70 560h}$, show that the general solution is given by $h=17 640{\left(c-R^{2}t\right)}^2$, where $c\in \mathrm{ℝ}$.

Show that $\frac{dH}{dt}\approx 0.2514-0.009873t-0.1405\sqrt{H}$, where $0\le t\le T$.

Use the general solution from part (d) and the initial condition $h\left(0\right)=3.2$ to predict the value of $T$.

Find this new height.

Use Euler’s method with a step length of $0.5$ minutes to estimate the maximum value of $H$.

Show that $\frac{dh}{dt}=-R^2\sqrt{70 560h}$.

From your solution to part (a), or otherwise, find the terminal velocity of the object predicted by this model.

Use the differential equation to find the terminal velocity for the object.

Use Euler’s method, with a step length of 0.2, to find the displacement and velocity of the object when $t=0.6$.

By repeated application of Euler’s method, find an approximation for the terminal velocity, to five significant figures.

Use your answers to parts (d), (e) and (f) to comment on the accuracy of the Euler approximation to this model.

Write down the differential equation as a system of first order differential equations.

Find the general solution of the differential equation $Q\text{'}\left(t\right)=\beta NQ\left(t\right)$.

By substituting $v=\frac{\text{d}x}{\text{d}t}$ into the equation, find an expression for the velocity of the particle at time $t$. Give your answer in the form $v=f(t)$.

An earth satellite moves in a path that can be described by the curve $72.5x^2+71.5y^2=1$ where $x=x(t)$ and $y=y(t)$ are in thousands of kilometres and $t$ is time in seconds.

Given that $\frac{\text{d}x}{\text{d}t}=7.75\times {10}^{-5}$ when $x=3.2\times {10}^{-3}$, find the possible values of $\frac{\text{d}y}{\text{d}t}$.

Give your answers in standard form.