Firstly, let's answer yesterday's questions:
You are given a box, measuring 6 feet by 6 feet by 6 feet. A simple first observation is that the box (which is actually a big cube) has a volume of 216 square feet in total. Secondly, using the Pythagorean Theorem (each side of the box is 6 feet), the diameter across the box would be the square root of 36 + 36, or the square root of 72, which is 8.48 feet.
1. What would be the diameter of each of four equally-sized spheres which would fit snugly into the box? Each sphere could only have a diameter of 4.24 feet (because the total diameter width of the box is 8.48 feet)...two spheres, packed side by side, would have a combined diameter of 8.48 feet. Each sphere could have a maximum diameter of 4.24 feet if the balls were packed as snugly as possible (e.g. across the diameter of the box)
2. How much excess (wasted) space would be left in the box after it was snugly packed with the spheres? The formula for the volume of a sphere is 4/3 x Pi x r cubed, where Pi = 22/7, or approximately 3.14 (a constant) and r = the radius of the sphere (which is one-half of the diameter of the sphere, or 2.12 feet). Substituting, we get 4/3 x 3.14 x 2.12 x 2.12 x 2.12 = 39.91cubic feet. The volume of space contained within each of the four spheres is 39.91 cubic feet. The volume contained within all of the four spheres together is then 39.91 x 4 = 159.64 cubic feet.
Since the total box contains 216 cubic feet, and the total of four spheres takes up 159.64 cubic feet, the difference (or the excess space) is 216 - 159.64 = 56.36 cubic feet of "wasted" space.
3. How much excess (wasted) space would be left in the box if it were instead snugly packed with eight equally-sized spheres? Before using the same reasoning as in questions 1 and 2, above, you must first picture how the eight balls would be fitted into the container. This will require some very creative visualization as to how exactly the balls must be positioned. Take the weekend to think about it.
For today, I offer you some tongue twisters (to give your mind a break from the closely=packed spheres problem, above:
*Betty Botter had some butter,"But," she said, "this butter's bitter. If I bake this bitter butter, it would make my batter bitter. But a bit of better butter--that would make my batter better."So she bought a bit of butter, better than her bitter butter, and she baked it in her batter, and the batter was not bitter. So 'twas better Betty Botter bought a bit of better butter.
*Six thick thistle sticks. Six thick thistles stick.
*Is this your sister's sixth zither, sir?
*A big black bug bit a big black bear,made the big black bear bleed blood.
*The sixth sick sheik's sixth sheep's sick.
*Toy boat. Toy boat. Toy boat.
Have a wonderful weekend. Try not to spit on your computer screen while you try and test those tricky tongue twisters time after time.