3.4 Radians, arcs and sectors questions for IB Maths AA

Find the value of $\theta$.

Find the exact area of the sector $\mathrm{A}\mathrm{O}\mathrm{B}$.

Find the exact volume of the space enclosed by the hat and its circular base.

Express $h$ in terms of $r$.

Let $A$ cm${}^2$ be the area of paper used to make the cup. Show that $A=\sqrt{{\pi }^2{r}^4+{\displaystyle \frac{562500}{r^2}}}$.

Find the value of $r$ that minimises $A$, and the minimum value of $A$.

For the cup with minimum area, find the angle between the slanting side of the cup and its vertical axis.

Each cup is cut from a sector of a circle of paper and the two straight edges are joined without overlap. For the cup with minimum area, find the angle of the sector, in degrees.

Show that $A_n={\displaystyle \frac{n}{2}}\sin \left({\displaystyle \frac{2\pi }{n}}\right)$.

Show that $p_n=n\sin \left({\displaystyle \frac{\pi }{n}}\right)$.

Find the exact values of $p_6$, $A_6$ and $A_{12}$.

Use your GDC to find the values of $p_{24}$ and $A_{24}$, giving your answers correct to five significant figures.

By writing $x={\displaystyle \frac{1}{n}}$, use l'Hôpital's rule to show that ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{p}_n=\pi }$.

Find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{A}_n}$.

Use l'Hôpital's rule to show that ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{n}^2(\pi -p_n)={\displaystyle \frac{{\pi }^3}{6}}}$.

It can be shown that ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{n}^2(\pi -A_n)={\displaystyle \frac{2{\pi }^3}{3}}}$.

Let $c_n=\lambda p_n+(1-\lambda )A_n$, where $\lambda \in \mathbb{R}$. Find the value of $\lambda$ for which ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{n}^2(\pi -c_n)=0}$.

For this value of $\lambda$, find the exact value of $c_6$.

With $\lambda$ equal to the value found in part (h), find the least value of $n$ for which $\pi -p_n<{10}^{-4}$, and the least value of $n$ for which $\pi -c_n<{10}^{-4}$.

Express ${135}^{\circ }$ in radians, as an exact multiple of $\pi$.

Find the exact length of the arc of the sector.

Find the exact area of the sector.

A second sector of the same circle has area ${\displaystyle \frac{64\pi }{5}}$ cm${}^2$. Find the angle at the centre of this sector, giving your answer in degrees.

Find the length of the chord [AB].

Find the area of the minor segment cut off by the chord [AB].

Find the value of $r$.

Find the perimeter of the sector.