GK 2018 Q21
21. Freda's flying club designed a flag of a flying dove on a square grid as shown.
The area of the dove is 192 cm2. All parts of the perimeter of the dove are either quarter-circles or straight lines. What are the dimensions of the flag?
- A 6 cm × 4 cm
- B 12 cm × 8 cm
- C 21 cm × 14 cm
- D 24 cm × 16 cm
- E 27 cm × 18 cm
Grey Kangaroo 2018, Question 21. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 24 cm × 16 cm
Let each of the small squares in the grid have side-length \(x\) cm. Remove the shading and divide the dove into regions as shown.
It can be seen that the regions marked A and B combine to make a square of side \(2x\) cm and hence of area \(4x^2\) cm2. Similarly, regions C, D and E combine to make a rectangle with sides \(2x\) cm and \(3x\) cm and hence area \(6x^2\) cm2. Finally, region F is a rectangle with sides \(2x\) cm and \(x\) cm and hence area \(2x^2\) cm2. Since the total area of the dove is 192 cm2, we have \(4x^2 + 6x^2 + 2x^2 = 192\) and hence \(12x^2 = 192\). Therefore \(x^2 = 16\) and hence \(x = 4\). Hence the flag has length (6 × 4) cm = 24 cm and height (4 × 4) cm = 16 cm.
Solution © UK Mathematics Trust, from the GK 2018 solutions.