• Dec 26th 2007, 12:35 AM
geton

In the diagram, http://farm3.static.flickr.com/2255/...474b2ea3_m.jpg
AD and BC are arcs of circles with centre O, such that OA = OD = r cm, AB = DC = 8 cm and angle BOC = θ radians.

Given that the area of the shaded region is 48 cm^2, show that r = 6/θ – 4.
• Dec 26th 2007, 03:04 AM
Soroban
Hello, geton!

Are you familiar with the formula for the area of a sector? .$\displaystyle A \;=\;\frac{1}{2}r^2\theta$

Quote:

$\displaystyle \text{In the diagram: }AD\text{ and }BC\text{ are arcs of circles with centre }O,$
$\displaystyle \text{such that: }\:OA = OD = r,\;AB = DC = 8,\;\text{ and }\;\angle BOC = \theta\text{ radians.}$

$\displaystyle \text{Given that the area of the shaded region is }48\text{ cm}^2\text{, show that: }\: r \:= \:\frac{6}{\theta}- 4$

The area of sector $\displaystyle AOB \:=\:\frac{1}{2}r^2\theta$

The area of sector $\displaystyle BOC \:=\:\frac{1}{2}(r+8)^2\theta$

Their difference is: .$\displaystyle A \:=\:\frac{1}{2}(r+8)^2\theta - \frac{1}{2}r^2\theta \:=\:48$

. . . . . Solve for $\displaystyle r.$