IMC 2011 Q8
8. A square piece of card has a square of side 2 cm cut out from each of its corners. The remaining card is then folded along the dotted lines shown to form an open box whose total internal surface area is 180 cm2.
What is the volume of the open box in cm3?
- A 100
- B 128
- C 162
- D 180
- E 200
Intermediate Mathematical Challenge 2011, Question 8. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 200
The base of the open box is a square. Let its side be of length \(x\) cm. Then the total surface area of the box in cm2 is \(x^2 + 4 \times 2x = x^2 + 8x\). Hence \(x^2 + 8x = 180\), that is \(x^2 + 8x - 180 = 0\). Therefore \((x + 18)(x - 10) = 0\), which gives \(x = -18\) or \(x = 10\). As \(x\) is positive, it may be deduced that the open box has dimensions 10 cm × 10 cm × 2 cm. So its volume is 200 cm3.
Solution © UK Mathematics Trust, from the IMC 2011 solutions. Longer versions with investigations: Solutions and investigations (PDF).