Review question

# If we know the difference in their areas, how big are these sectors? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9423

## Solution

The figure shows two sectors in which $AB$, $CD$ are arcs of concentric circles, centre $O$ and radii $\quantity{2k}{cm}$ and $\quantity{3k}{cm}$ respectively. The angle $A\hat{O}B$ is $\dfrac{3}{4}$ radian and the area $ABDC$ is $\quantity{30}{cm^2}$. Find

1. the value of $k$,

The area of a sector with radius $r$ containing an angle $\theta$ is $A=\dfrac{1}{2}r^2\theta$. Hence the difference in the sector areas $30 = \dfrac{1}{2}(3k)^2\dfrac{3}{4} - \dfrac{1}{2}(2k)^2\dfrac{3}{4}$.

Solving this for $k$ gives $k=4$.

Find

1. the difference between the lengths of the arcs $AB$ and $CD$.

The length of an arc of radius $r$ and given by an angle $\theta$ is $r\theta$. Hence the difference between arc $AB$ and arc $CD$ is $12\times\dfrac{3}{4} - 8\times\dfrac{3}{4} = \quantity{3}{cm}$.