IB Maths AI HL Sl 3 4 Arc Length and Sector Area Questions

Practise converting angles and using arc-length or sector-area formulae to find unknown dimensions, checking units, diagrams and rounding.

Exam points

Convert between degrees and circle fractions as needed to calculate arc length or sector area using the given angle.
Apply arc-length and sector-area formulae to determine an unknown radius, angle, length or area.
Interpret the result in a diagram or applied context with correct units and rounding.

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Study SL 3.4—Arc length and sector area conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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The following question compares the distance and direction between cities on a flat surface to the distance and direction between cities on a sphere.
Consider a model where the cities of Bogotá, Moscow, and Nairobi lie on a flat surface. In this model, Nairobi is 6000 km due south of Moscow and Bogotá is 12500 km due west of Nairobi, as shown in the following diagram.

Figure for Question IB Maths AI HL Sl 3 4 Arc Length and Sector Area Questions question 1 — IB Maths AI HL

Show that the distance between P and N along the arc from P to N is $3000 \pi \mathrm{~km}$ .

Point A, which is also on the equator, has position vector $\boldsymbol{a}=\left(\begin{array}{l}6 \\ 0 \\ 0\end{array}\right)$ as shown in the
following diagram.

P, N and A , and the arcs connecting them, form a spherical triangle.
The angle at vertex A is defined as the angle between the vectors $\boldsymbol{a} \times \boldsymbol{p}$ and $\boldsymbol{a} \times \boldsymbol{n}$ .

attempt to use arc length formula $s=r \theta$ OR $\frac{90^{\circ}}{360}$ of circumference
$=6000\left(\frac{\pi}{2}\right)$ OR $\frac{1}{4} \times 2 \pi \times 6000$ $=3 \pi$ thousand $\mathrm{km} \quad$ OR $\quad 3000 \pi \mathrm{~km}$

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IB Maths AI HL Sl 3 4 Arc Length and Sector Area Questions question 1