3.7 Radian measure questions for IB Maths AI

Find the value of $a$ and the value of $d$.

Find the value of $b$.

Find the value of $c$.

Write down, in radians, the angle at the centre of the wheel between two adjacent capsules. Give your answer as an exact multiple of $\pi$.

Find the distance between two adjacent capsules, measured along the rim.

Find the straight-line distance between two adjacent capsules.

Write down the angular speed of the wheel in radians per minute.

Find the distance travelled by a capsule in 4 minutes.

Valentina boards a capsule at the lowest point of the wheel at time $t=0$. Her height above the ground, $h$ metres, $t$ minutes later is modelled by

Find Valentina's height after 4 minutes.

Find the length of time, during one rotation, for which Valentina is more than 70 m above the ground. Give your answer in minutes.

Write down $V_1$ and $V_2$ each as the imaginary part of a complex function of the form $a{\mathrm{e}}^{\mathrm{i}(100\pi t+c)}$.

Show that $V_1+V_2=\mathrm{I}\mathrm{m}\left(z{\mathrm{e}}^{\mathrm{i}100\pi t}\right)$, where $z=30+40{\mathrm{e}}^{\mathrm{i}\frac{\pi }{3}}$.

Find $z$ in the form $a+b\mathrm{i}$, giving the exact values of $a$ and $b$.

Hence write $V_1+V_2$ in the form $A\sin (100\pi t+\phi )$, where $A>0$ and $0<\phi <\frac{\pi }{2}$.

Find the first time $t>0$ at which the total voltage $V_1+V_2$ is a maximum. Give your answer in milliseconds.

The total voltage is applied to a circuit in which a resistor, an inductor and a capacitor are connected in series. Each component has a complex impedance:

• a resistor of $R$ ohms has impedance $R$

• an inductor of $L$ henrys has impedance $\mathrm{i}\omega L$

• a capacitor of $C$ farads has impedance $\frac{1}{\mathrm{i}\omega C}$

where $\omega$ is the angular frequency. For components in series, the total impedance $Z$ is the sum of their impedances. In Kwame's circuit $R=20$, $L=0.15$ and $C=100\times {10}^{-6}$.

Find $Z$ in the form $a+b\mathrm{i}$.

Find the modulus and the argument of $Z$.

The current in the circuit, $I$ amperes, is given by $I=\mathrm{I}\mathrm{m}\left(w{\mathrm{e}}^{\mathrm{i}100\pi t}\right)$, where $w=\frac{z}{Z}$.

Find $I$ in the form $B\sin (100\pi t+\psi )$, where $B>0$.

Find the time interval between a maximum of the voltage and the next maximum of the current. Give your answer in milliseconds.

The circuit is said to be at resonance when its total impedance is a real number.

Kwame can change the angular frequency of the supply. Find the frequency, $f=\frac{\omega }{2\pi }$ hertz, at which this circuit is at resonance.

Kwame now uses the original supply with only the resistor and the inductor, connected in parallel. For two components in parallel, the total impedance $Z_{\text{p}}$ satisfies

Find $Z_{\text{p}}$ in the form $a+b\mathrm{i}$.

Hence find the amplitude of the current in this parallel circuit.

State one reason why the current measured by Kwame might differ from the value predicted by this model.

Show that the period of the model is 365 days, correct to three significant figures.

The model can also be written as $D(t)=7.2\sin (q(t-80))+12.3$, where $t-80$ is measured in radians. Find the value of $q$.

Find the number of days in the year on which there are more than 16 hours of daylight.

Find the area of lawn watered.

Find the perimeter of the watered region.

Write down the value of $\theta$ in degrees.

Inés now plans a new flower bed in the shape of a sector with radius $r$ m and angle $\theta$ radians. The area of the flower bed must be $90{\text{ m}}^2$.

Show that $\theta =\frac{180}{r^2}$.

The whole perimeter of the flower bed will be edged with metal strip. Show that the length of strip needed, $P$ m, is given by

Find $\frac{\mathrm{d}P}{\mathrm{d}r}$.

Hence find the value of $r$ that minimizes $P$, and the minimum length of strip needed.

Find the corresponding value of $\theta$.

The metal strip costs 14.50 euros (EUR) per metre. Find the minimum cost of the strip. Give your answer correct to two decimal places.

Write down the angle at the centre in radians, as an exact multiple of $\pi$.

Find the length of the curved edge of the flower bed.

Find the area of the flower bed.