Stacking Footballs

1. Stacking Footballs How many footballs can you fit in a classroom?

2. 8m Room 26 3m 8m r = 0.1m Initial information • Room 26 is 8m long by 8m wide by 3m tall • A football has a diameter of 0.2m so a radius of 0.1m

3. Volume of classroom What is wrong? Space is always wasted so you cannot ever fit that many in! Method 1 • Volume of football

4. 0.2m 0.2m Method 2 – box stacking > box size • Each football fits in a box.

5. Number of balls in room How many balls (in boxes) fit along each side? 8m 3m 8m Method 2- box stacking > putting it together • What is wrong? • Wastes half the space!

6. 0.2m x 0.1m Method 3 – orange stacked block > height • Using Pythagoras’ Theorem:

7. We now have the basis for a vertical line: Using this line we work out how many footballs high the orange stacked block is  0.173m 0.173m 0.1m Method 3 – orange stacked block > height

8. Rule for height: On the diagram: Method 3 – orange stacked block > height

9. We already know that 40 balls fit in the width of the back wall. However, alternate rows will only be 39 balls wide so that the rows fit together. Therefore in 17 rows there will be: = 672 balls per orange stacked block One block is as wide as one ball so 40 blocks fit along the length of the room Method 3 – orange stacked block > width

10. 672 balls per block x40 block = 26880 balls fit in the room using method 3 What is wrong? Space wasted between blocks Method 3 – orange stacked block > evaluation

11. The room sideways on As before, we can calculate x to be 0.173m 0.2m 0.1m x Method 4 – complete orange stacked > width

12. Rule for number of blocks (width) = So the room will contain 46 rows of orange stacked blocks However, row 2 must fit into row1 so cannot have the same number of balls in it Method 4 – complete orange stacked > width

13. Block 2 In block 2, one row must be missed out to allow block 2 to fit into block 1 Block 1 contains 17 rows so block 2 contains 16 rows Method 4 – complete orange stacked > block 2

14. The room sideways on Room is 46 blocks wide = Method 4 – complete orange stacked > putting it together + • Total balls = 14536 + 15456= 299992 balls

15. This is the best answer as complete orange packing is proved to be the most efficient way (Kepler) What is wrong? The value of 0.173 was rounded to 3DP so we could round the answer to 30000 balls However, you cannot have 16.5 rows so rounding is accurate. 29992 should be an accurate answer Method 4 – complete orange stacked > evaluation