Answer:
Both boxes have the same amount of empty space
Explanation:
Given
Cube Box
![Length = 10cm](https://tex.z-dn.net/?f=Length%20%3D%2010cm)
Robert:
1 glass marble
![Diameter = 10cm](https://tex.z-dn.net/?f=Diameter%20%3D%2010cm)
Susan:
1000 glass marble
![Diameter = 1cm](https://tex.z-dn.net/?f=Diameter%20%3D%201cm)
Required
Whose box has more empty space?
First, we need to calculate the volume of the cubic box.
![Volume = Length^3](https://tex.z-dn.net/?f=Volume%20%3D%20Length%5E3)
Substitute 10cm for Length
![Volume = 10^3](https://tex.z-dn.net/?f=Volume%20%3D%2010%5E3)
![Volume = 1000cm^3](https://tex.z-dn.net/?f=Volume%20%3D%201000cm%5E3)
Next, calculate the volume of Robert's glass
Volume is calculated as:
---- Volume of a sphere
Where
r = radius
![r = \frac{1}{2} * Diameter](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%2A%20Diameter)
--- Given
Substitute 10 for diameter
![r = \frac{1}{2} * 10cm](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%2A%2010cm)
![r = 5cm](https://tex.z-dn.net/?f=r%20%3D%205cm)
So:
![Volume = \frac{4}{3}\pi r^3](https://tex.z-dn.net/?f=Volume%20%3D%20%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3)
![Volume = \frac{4}{3} * \frac{22}{7} * 5^3](https://tex.z-dn.net/?f=Volume%20%3D%20%5Cfrac%7B4%7D%7B3%7D%20%2A%20%5Cfrac%7B22%7D%7B7%7D%20%2A%205%5E3)
![Volume = \frac{4}{3} * \frac{22}{7} * 125](https://tex.z-dn.net/?f=Volume%20%3D%20%5Cfrac%7B4%7D%7B3%7D%20%2A%20%5Cfrac%7B22%7D%7B7%7D%20%2A%20125)
![Volume = 523.8cm^3](https://tex.z-dn.net/?f=Volume%20%3D%20523.8cm%5E3)
Next, we determine the volume of Susan's 1000 glasses
Volume is calculated as:
---- Volume of a sphere
Where
r = radius
![r = \frac{1}{2} * Diameter](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%2A%20Diameter)
--- Given
Substitute 1cm for diameter
![r = \frac{1}{2} * 1cm](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%2A%201cm)
![r = 0.5cm](https://tex.z-dn.net/?f=r%20%3D%200.5cm)
Substitute 0.5cm for radius in ![Volume = \frac{4}{3}\pi r^3](https://tex.z-dn.net/?f=Volume%20%3D%20%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3)
![Volume = \frac{4}{3} * \frac{22}{7} * 0.5^3](https://tex.z-dn.net/?f=Volume%20%3D%20%5Cfrac%7B4%7D%7B3%7D%20%2A%20%5Cfrac%7B22%7D%7B7%7D%20%2A%200.5%5E3)
![Volume = \frac{4}{3} * \frac{22}{7} * 0.125](https://tex.z-dn.net/?f=Volume%20%3D%20%5Cfrac%7B4%7D%7B3%7D%20%2A%20%5Cfrac%7B22%7D%7B7%7D%20%2A%200.125)
![Volume = 0.5238](https://tex.z-dn.net/?f=Volume%20%3D%200.5238)
But there are 1000 glasses.
So, the volume of 1000 glasses is:
![Volume = 0.5238 * 1000](https://tex.z-dn.net/?f=Volume%20%3D%200.5238%20%2A%201000)
![Volume = 523.8cm^3](https://tex.z-dn.net/?f=Volume%20%3D%20523.8cm%5E3)
--------------------------------------------------------------------------------------------------------------
For Robert:
![Volume = 523.8cm^3](https://tex.z-dn.net/?f=Volume%20%3D%20523.8cm%5E3)
For Susan:
![Volume = 523.8cm^3](https://tex.z-dn.net/?f=Volume%20%3D%20523.8cm%5E3)
Since, they have the same volume.
Both boxes have the same amount of empty space