C hristmas Problem Solving GCSE Shapes and Space

Transcription

1 C hristmas Problem Solving GCSE Shapes and Space Question 1 Ahmed has a roll of wrapping paper 2m long and 75cm wide. He has to wrap the following presents, does he have enough paper? Show your workings. HINT The maze-in-a-ball can be wrapped with a rectangular piece of paper Maze-in-a-ball Speakers X2 10cm 12.5cm SPHERE Radius 10cm CUBES Side length cm each Pyramid of chocolate Bubble bath gift set 22.5cm 30cm 27cm SQUARE BASED RIGHT PYRAMID Square base side length 27cm Slant height 22.5cm 23cm TRIANGULAR PRISM Triangle base (equilateral) side length 23cm Prism height 30cm Does Ahmed have enough paper to wrap each present in a single sheet (of any shape)?

2 Question 2 Julie is putting 3 presents a jumper, a pair of socks and a teddy bear into 3 gift boxes. The boxes are cubes of different sizes. Box 1 Box 2 Box 3 x cm y cm z cm Use the information in the table to answer the questions: Volume of object Jumper 11,000 cm 3 Teddy Bear 50,000 cm 3 Socks 750 cm 3 Clues Box 1 can hold the socks but nothing else. Box 2. There are three ways to select two presents. Two of these combinations fit into Box 2. Box 3 can hold all three items without being full but could not hold an extra jumper in addition to the first three items. 2a) For each box write down an inequality to describe the range of values the volume of the box could take. 2b) Use your answer to part 1 to write down an inequality describing the range of possible values of: i) x ii) y iii) z Question 3 Megan and her Dad have bought a Christmas tree and want to take it home in the boot of their car. The tree is 170cm tall. First they assume that the car boot is a cuboid. Will the tree fit? Show your working. 120 cm 73 cm 106 cm The wheel arches of the car mean that the boot is actually a trapezoidal prism. As the wheel arches take up 5cm each side, will the Christmas tree still fit? View from above 120 cm 120 cm 73 cm 106 cm 5 cm 5 cm 5 cm

3 Question 4 When Megan and her dad get the Christmas tree home, they need to rearrange their furniture to fit it in the living room. Use the clues to place all the furniture into the room. x2 tables s Lamp Television Foot stool Clues: The rug has 3 whole objects on top of it, including the dog Other than what is on the rug, no objects touch each other Objects can be against the wall, but must not touch the window or the radiator There must be enough room to open and close the door Tree Rug 4a) When all the furniture is in the room estimate how much floor has been covered. Give your answer in units 2. Dog 1 unit 2 Window Radiator Doorway Exactly how much of the floor has been covered?

4 Question 1 Solutions Yes all the objects can be wrapped with a single roll of paper (200cm x 75cm). Present Shape of paper Surface area of wrapping paper (Rounded up to 1 decimal place) Maze-in-a-ball Rectangle cm cm = 1974cm² Speakers Cube net X cm² = 937.5cm² EACH (or cm² if wrapped together) Pyramid of chocolate Pyramid net 27² + 4 (½ ) = 1944cm² Bubble bath gift set Triangular prism net 2 (½ ) + 3 (23 30) = cm² Total = cm² (or cm²) The sheets of paper to cover each present can be cut from the roll of paper as below* Maze-in-a-ball Pyramid of chocolate Bubble bath gift set Speakers * Not to scale Question 2 2a 750 cm 3 volume of box 1 < cm cm 3 volume of box 2 < cm cm 3 < volume of box 3 < cm 3 2b 9.09 x < y < < z < 41.75

5 Question = = ( = = 160 3sf) ( 25636) 2 = = = = 175cm (rounded down to nearest cm) Yes, the tree will fit = = ( = = 156 3sf) ( 24461) 2 = = = = 172cm (rounded down to nearest cm) Yes, the tree will still fit. Question 4 There are multiple solutions. One solution is shown here: In this solution, the exact amount of floor covered is π units 2 Lamp Tree Foot stool Television Dog table Rug Window table Radiator Doorway