184. Overlapping sectors


In the diagram is rectangle ABCD with height 10 cm.  An arc with centre at point B is drawn from point A to side BC.  An arc with centre at point C is drawn from point D to side BC.  Given that the shaded (coloured) regions ⓐ and ⓑ have equal area, determine the length of BC.


183. A matter of threes


Α, B and C are three numbers.  When Α has 3 added to it, B has 33 subtracted from it, and C has 3 divided into it, the resultant numbers are equal.  The sum of Α, B and C is 333.  Determine the value of Α.


182. Cube slice

In the diagram, three points P, Q and R lie on a plane that cuts the cube into two sections.  Also, points P, Q and R are midpoints of their respective edges.

⒜  What is the shape of the surface formed by the cut?  Also, determine the most appropriate name for the smaller section.
⒝* Determine the ratio of the volume of the larger section to the volume of the smaller section.


181. Water droplets



The rate of melting of ice or evaporation of a liquid is a function of surface area.  

A spherical water droplet has volume 1 ml.  Two other spherical water droplets each have volume 0.5 ml.  Determine which—the single big droplet or the two smaller droplets—has a greater total surface area, and how much greater (as a ratio or percentage).