Answer:
About 280 km
Explanation:
1. Find the footprint area of a marble
A marble will fit into a square 1.6 cm on each side.
A = l² = (1.6 cm)² = 2.56 cm²
2. Convert the area of the continental U.S. into square centimetres
(a) Convert to square metres
[tex]A = 8.08 \times 10^{6} \text{ km}^{2}\times \left(\dfrac{\text{1000 m}}{ \text{1 km}}\right) ^{2} = 8.08 \times 10^{12}\text{ m}^{2}[/tex]
(b) Convert to square centimetres
[tex]A = 8.08 \times 10^{12} \text{ m}^{2}\times \left(\dfrac{\text{100 cm}}{ \text{1 m}}\right) ^{2} = 8.08 \times 10^{16}\text{ cm}^{2}[/tex]
3. Calculate the number of marbles to make one layer.
[tex]\text{No. of marbles} = 8.08 \times 10^{16} \text{ cm}^{2}\times \left(\dfrac{ \text{1 marble }}{ \text{2.56 cm}^{2}}\right) = 3.12 \times 10^{16}\text{ marbles}[/tex]
4. Calculate the number of layers needed for Avogadro's number of marbles
Assume the marbles will stack on top of each other like sugar cubes.
[tex]\text{No. of layers}\\= 6.022 \times 10^{23} \text{ marbles} \times \left(\dfrac{ \text{1 layer}}{3.16 \times 10^{16}\text{ marbles}}\right) =1.91\times 10^{7} \text{ layers}[/tex]
5. Calculate the height of the layers
[tex]h = 1.91 \times 10^{7} \text{ layers} \times \dfrac{\text{1.6 cm}}{\text{1 layer}} \times \dfrac{\text{1 cm}}{\text{100 m}} \times \dfrac{\text{1 km}}{\text{1000 m}} = \text{305 km}[/tex]
However, the marbles aren't cubes; they are spheres. Each layer of marbles will slide into the "dimples" of the layer below, like packing oranges into a crate.
The effective height of each layer decreases by about 10 %.
The height of the stack will be about 280 km. That's approximately the straight-line distance from Boston to New York.
