Thinking Outside the box 259 To have a landing pad of size at least 1 m x 3 m, we need two of these boxes lengthwise and five widthwise, for a total of 10 boxes per layer. We determine the total height h1 of the pile as follows The average height of an adult male African elephant is 3. 2 m [Theroux 2002]; the motorcyclist could easily clear such an elephant with a jump of ho a 4 m. From this value, and(4), we obtain 6 thus, we want h1 N 0.67m= 26. 2 in. To exceed this number with 6-inch layers, we need 5 lavers To summarize: We use 6in x 28 in x 28 in boxes. There are 10 boxes in each layer of the stack, in a 2 x 5 grid, and the stack consists of 5 layers of boxes with 36 layers of cross-hatched cardboard flats piled on top Changing the Parameters Because the dimensions of the boxes used vary as the cube root of b which is ther linear or inversely linear in most of our parameters, our results are fairly resistant to change in any parameter. For example, changing one of them by factor of 2 changes the optimal box dimensions by only 25%; even increasing a parameter by an order of magnitude only doubles the dimen The one exception is the jumping height. Since B is independent of that, so are the optimum box dimensions, except insofar as the height affects the amount of area. However, the jumping height does affect the height of the box pile; we want that to increase linearly with the jumping height, in a ratio given by the desired deceleration g. Additionally, a very high or very low jumping height will cause our model to break down completely. For C-flute cardboard in the flats, the total amount of cardboard needed essentially depends only on the weight of the motorcycle and rider and on the net deceleration. This may at first glance appear counterintuitive: Should we not expect the jumping height to affect the stress put on the flats? In fact, the height is irrelevant as long as the assumptions of the model are justified. Since the boxes below the flats are calculated to break upon experiencing a force mg, the force transmitted through the flats by the motorcyclist is never larger than this. The only exception is the initial force that the flats experience on first being hit by the motorcycle. If the jumping height is sufficiently large, the assumption-that this initial force is dominated by the normal force exerted by the boxes underneath-will break down, and we may need to increase the thickness beyond the value calculated. However, forjumping heights this large, it is probable that other parts of our model will break down Certain predictions of our model are independent of its parameters, so long as our assumptions are justified. Most notable is the observation that to best conserve material for a given result, the height of a box should be one-quarterThinking Outside the Box 259 To have a landing pad of size at least 1 m × 3 m, we need two of these boxes lengthwise and five widthwise, for a total of 10 boxes per layer. We determine the total height h1 of the pile as follows The average height of an adult male African elephant is 3.2 m [Theroux 2002]; the motorcyclist could easily clear such an elephant with a jump of h0 ≈ 4 m. From this value, and (4), we obtain h1 = h0g g + g = h0 6 ; thus, we want h1 ≈ 0.67 m = 26.2 in. To exceed this number with 6-inch layers, we need 5 layers. To summarize: We use 6 in × 28 in × 28 in boxes. There are 10 boxes in each layer of the stack, in a 2×5 grid, and the stack consists of 5 layers of boxes with 36 layers of cross-hatched cardboard flats piled on top. Changing the Parameters Because the dimensions of the boxes used vary as the cube root of B, which is either linear or inversely linear in most of our parameters, our results are fairly resistant to change in any parameter. For example, changing one of them by a factor of 2 changes the optimal box dimensions by only 25%; even increasing a parameter by an order of magnitude only doubles the dimensions. The one exception is the jumping height. Since B is independent of that, so are the optimum box dimensions, except insofar as the height affects the amount of area. However, the jumping height does affect the height of the box pile; we want that to increase linearly with the jumping height, in a ratio given by the desired deceleration g . Additionally, a very high or very low jumping height will cause our model to break down completely. For C-flute cardboard in the flats, the total amount of cardboard needed essentially depends only on the weight of the motorcycle and rider and on the net deceleration. This may at first glance appear counterintuitive: Should we not expect the jumping height to affect the stress put on the flats? In fact, the height is irrelevant as long as the assumptions of the model are justified. Since the boxes below the flats are calculated to break upon experiencing a force mg , the force transmitted through the flats by the motorcyclist is never larger than this. The only exception is the initial force that the flats experience on first being hit by the motorcycle. If the jumping height is sufficiently large, the assumption—that this initial force is dominated by the normal force exerted by the boxes underneath—will break down, and we may need to increase the thickness beyond the value calculated. However, for jumping heights this large, it is probable that other parts of our model will break down. Certain predictions of our model are independent of its parameters, so long as our assumptions are justified. Most notable is the observation that to best conserve material for a given result, the height of a box should be one-quarter