A pattern on a T-shirt, shown in Fig. 1, consists of a sequence of similar sectors. The first sector \(AOB\) of a circle has centre \(O\), and radii \(OA\) and \(OB\).
The angle \(AOB\) is 1.2 radians. Given that the maximum area of \(AOB\) is \(57\text{ cm}^2\), find the range of values of \(OA\).[2]
[The area of a sector of a circle with radius \(r\) is \(\frac12r^2\theta\), where \(\theta\) is the angle, in radians, subtended by the arc at the centre.]
The area of \(AOB\) is now fixed at \(60\text{ cm}^2\), and the area of each sector is \(\frac35\) of the area of the preceding one.
Find the area of the 5th sector in the pattern.[2]
Given that the total area of \(n\) sectors printed cannot exceed \(148.48\text{ cm}^2\), find the maximum number of sectors that can be printed.[3]
The cost of printing the first sector \(AOB\) is \(\$C\) and the cost of printing each sector is \(\$0.15\) less than the preceding one. Find the maximum value of \(C\) if the total cost of printing the first 5 sectors does not exceed \(\$4.99\).[2]