IMC 2016 Q17
17. The shaded region in the diagram, bounded by two concentric circles, is called an annulus. The circles have radii 2 cm and 14 cm.
The dashed circle divides the area of this annulus into two equal areas.
What is its radius?
- A 9 cm
- B 10 cm
- C 11 cm
- D 12 cm
- E 13 cm
Intermediate Mathematical Challenge 2016, Question 17. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 10 cm
Let the radius of the dashed circle be \(r\) cm. Then one of the equal areas is bounded by circles of radii of 14 cm and \(r\) cm, whilst the other is bounded by circles of radii of \(r\) cm and 2 cm. So \(\pi \times 14^2 - \pi r^2 = \pi r^2 - \pi \times 2^2\). Dividing throughout by \(\pi\) gives \(196 - r^2 = r^2 - 4\). So \(2r^2 = 200\), that is \(r^2 = 100\). Therefore \(r = 10\) (since \(r > 0\)).
Solution © UK Mathematics Trust, from the IMC 2016 solutions. Longer versions with investigations: Solutions and investigations (PDF).