Standard +0.3 This is a straightforward application of sector area and arc length formulas. Students must set up an equation using the given shaded area (difference of two sectors), solve for the inner radius, then calculate the perimeter by adding two arc lengths and two radial segments. While it requires multiple steps, each step uses standard formulas with no conceptual difficulty or novel insight required.
\includegraphics{figure_3}
The diagram shows a sector of a circle, centre \(O\), where \(OB = OC = 15\) cm. The size of angle \(BOC\) is \(\frac{2}{5}\pi\) radians. Points \(A\) and \(D\) on the lines \(OB\) and \(OC\) respectively are joined by an arc \(AD\) of a circle with centre \(O\). The shaded region is bounded by the arcs \(AD\) and \(BC\) and by the straight lines \(AB\) and \(DC\). It is given that the area of the shaded region is \(\frac{90}{7}\pi\) cm\(^2\).
Find the perimeter of the shaded region. Give your answer in terms of \(\pi\). [5]
Question 3:
3 | Use correct sector area formula | M1
1 2 1 2 209
Obtain 152 π − x2 π = π or equivalent
2 5 2 5 5 | A1
=4
Obtain x | A1 | AWRT 4.00.
Use correct arc length formula twice | M1
38
Obtain 22+ π
5 | A1 | OE. Must be in terms of π.
Like terms must be collected.
Not from a rounded value of x.
5
Question | Answer | Marks | Guidance
\includegraphics{figure_3}
The diagram shows a sector of a circle, centre $O$, where $OB = OC = 15$ cm. The size of angle $BOC$ is $\frac{2}{5}\pi$ radians. Points $A$ and $D$ on the lines $OB$ and $OC$ respectively are joined by an arc $AD$ of a circle with centre $O$. The shaded region is bounded by the arcs $AD$ and $BC$ and by the straight lines $AB$ and $DC$. It is given that the area of the shaded region is $\frac{90}{7}\pi$ cm$^2$.
Find the perimeter of the shaded region. Give your answer in terms of $\pi$. [5]
\hfill \mbox{\textit{CAIE P1 2024 Q3 [5]}}