IB Mathematics Applications & Interpretation Topic…

A geologist models the decay of seismic wave amplitude $A$ with distance $d$ (in km) from an earthquake epicenter, using $A=k d^{n} e^{m d}$ , where $k, n, m$ are constants. Data are plotted as $

Graph of $\ln A$ against $\ln d$

$\ln A$ against $\ln d$ , where $\ln d$ means the natural logarithm of the numerical value of $d$ measured in km. For small $d$ , the term $e^{m d} \approx 1$ , so the plot is approximately a straight line on the axes passing through and .

Assuming $e^{m d} \approx 1$ , find the equation of the line in the form $\ln A=n \ln d+\ln k$ .

A pendulum's displacement, $x(t)$ , in meters, from its equilibrium position at time $t$ seconds is modeled by

x(t)=0.5 e^{-0.1 t} \cos \left(\frac{\pi}{2} t\right)

A drone's altitude, $A(t)$ , in meters, above a field at time $t$ seconds is modeled by

A (t) = 50 + 30 \sin (\frac{π}{20} t) + 20 e^{- 0.02}

A physicist models the decay of a radioactive substance, where the mass $M$ (in grams) remaining after time $t$ (in days) is given by $M=k e^{m t}$ , with constants. The physicist plots against and finds a straight line passing through and .

A wind turbine's power output, $P(t)$ , in kilowatts, varies over time $t$ hours according to the function

P (t) = 500 + 300 \sin (\frac{π}{12} t + \frac{π}{6}) +

Determine $k$ and $n$ .

If $m=-0.1$ , find an expression for $A$ in terms of $d$ .

Estimate $A$ when $d=2 \mathrm{~km}$ , to two significant figures.

Discuss the limitations of the model for large $d$ .

Find the initial displacement.

Determine the period of oscillation.

Show that the maximum displacement occurs at $t=0$ .

Find the time when the displacement is first at its equilibrium position.

The energy $E(t)$ of the pendulum is proportional to $x(t)^{2}$ . Given $E(0)=2$ , find the constant of proportionality and express .

Determine the rate of change of $E(t)$ at $t=1$ , correct to three significant figures.

Find the period of the drone's altitude oscillations.

Express the oscillatory part as a single trigonometric function with time-dependent amplitude.

Find the total energy consumed by the drone over the first 40 seconds, correct to two decimal places.

Determine the time when the drone first reaches an altitude of 60 meters.

Show that the relationship can be expressed as $\ln M=m t+\ln k$ .

Find the equation of the straight line in the form $\ln M=a t+c$ .

Determine the values of $k$ and $m$ .

Estimate the mass remaining after 10 days, to two decimal places.

Find the time when the mass is 10 grams, to the nearest day.

Determine the period of the oscillatory components of $P(t)$ .

Express the oscillatory part of $P(t), 300 \sin \left(\frac{\pi}{12} t+\frac{\pi}{6}\right)+200 e^{-0.05 t} \cos \left(\frac{\pi}{12} t\right)$ , in the form $A(t) \sin \left(\frac{\pi}{12} t+\phi(t)\right)$ , where $A(t)$ and $\phi(t)$ are functions of $t$ .

Sketch the graph of $P(t)$ for $0 \leq t \leq 24$ , indicating where $P(t)=600$ .

Find the time intervals in the first 24 hours when the turbine operates efficiently.

The cost of maintenance, $C(t)$ , in dollars per hour, is modeled as $C(t)=10+ 0.02 P(t)$ . Find the total maintenance cost over the first 24 hours, correct to two decimal places, using numerical integration.

IB Mathematics Applications & Interpretation Functions Questionbank

IB Mathematics Applications & Interpretation (AI) Topic Functions Questionbank