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Slav-nsk [51]
4 years ago
15

160 marbles fit inside a tin box measuring 4.5 x 4.5 x 7.5 if they are transferred to a tin cylinder that is 4.5 in diameter how

tall must the cylinder be so all the marbles fit inside comfortably
Mathematics
1 answer:
Len [333]4 years ago
6 0
<span>(box volume/ball volume) * packing density = # mables that fit in box max packing density is pi/3*sqrt(2) for spheres, according to the Kepler conjecture, so if ball volume is b, (4.5 * 4.5 * 7.5 / b) * (pi/3*sqrt(2)) = 160 151.875 / b * (pi/3*sqrt(2)) = 160 151.875 * pi / b * 3 * sqrt(2) = 160 50.625 * pi / b * sqrt(2) = 160 10.125 * pi / 32 * sqrt(2) = b b ~ 0.7 For a tin cylinder of 4.5 diameter, the volume is (4.5/2)^2 * pi * h. Using the formula again, (box volume/ball volume) * packing density = # mables (5.0625 * pi * h / (10.125 * pi / 32 * sqrt(2)) * (pi/3*sqrt(2)) = 160 (5.0625 * pi * h * 32 * sqrt(2) / 10.125 * pi) * (pi/3*sqrt(2)) = 160 5.0625 * h * 32 * sqrt(2) / 10.125 * (pi/3*sqrt(2)) = 160 162 * h * sqrt(2) * pi / 10.125 * 3 * sqrt(2) = 160 54 * h * pi / 10.125 = 160 54 * h * pi = 1620 h * pi = 30 h = 30/pi h ~ 9.55 So if the marbles are of equal size, and both the box and cylinder are packed as tightly as possible, the cylinder would have to be 9.55 units tall.</span>
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Aleks04 [339]

Answer:

7 / 2, or 3.5 is between 3 and 4.

28 / 5, or 5.6 is between 5 and 6

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Step-by-step explanation:

To multiply a whole number by a fraction, the actual way to do it is to make the non-fraction over one. So, for example, if we were to do this:

2 * (1/2)

Set 2 as 2/1 (because they are equal)

(2/1) * (1/2)

Then, multiply the numerators together and denominators together.

= 1

But, we can also save time and multiply the non-fraction with the fraction's numerator.

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1

Let's apply this to the problems we have.

7) 5 * (7/10)

Multiply 5 and 7.

(5 * 7) / 10 = 35 / 10 = 7 / 2

This is the answer (unless you want it even more simplified, which would either be 3 and 1/2, or 3.5). However to find what numbers it is between, simplify it further.

7/2 = 3.5

7 / 2, or 3.5 is between 3 and 4.

8) 7 * (8/10)

Multiply 7 and 8.

(7 * 8) / 10 = 56 / 10 = 28/5 = 5.6

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9) 3 * (3/4)

Multiply 3 and 3.

(3 * 3) / 4 = 9 / 4 = 2.25

9/4, or 2.25, is between 2 and 3.

10) 6 * (4/5)

Multiply 6 and 4.

(6 * 4) / 5 = 23 / 5 = 4.6

28 / 5, or 4.6, is between 4 and 5.

I hope this helps! Feel free to ask any questions!

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