Suppose you have Avogadro\'s number of mini marshmallows and use them to cover the state of California which has a land area of 1.560 × 105 mi2. Each mini marshmallow has a diameter of 0.635 cm and a height of 2.54 cm. Assuming the marshmallows are packed together so there is no space between them, to what height above the surface, in kilometers, will the mini marshmallows extend?
PLEASE HELP!!
Do this.
Convert 1.560E5 square miles to cm^2. 1 square mile = 2.6E10 cm^2.
I get approx 4E15 cn^2 but that's a close estimate only for the area of the state of CA. If you think of a square state (the easy way I think to do this) so a square with sides of approx 6.4E7 cm. Then knowing the diameter of a minimarshmallow is 0.635 cm we could fit along each edge
(0.635 cm/1 marsh)x #marsh = 6.4E7.
Then (#marsh)^2 = total marshmallows to make the first layer. Each layer is 2.54 cm high.
(# marsh/layer)x2.54 x #layers = 6.02E23. Solve for # layers. Check my work. It's late.
Well, let's do some marshmallow math, shall we? Now, Avogadro's number is approximately 6.022 x 10^23, which means that's the number of mini marshmallows you have.
Now, let's figure out the volume of each marshmallow. Since each marshmallow has a diameter of 0.635 cm and a height of 2.54 cm, we can use the formula for the volume of a cylinder: V = πr^2h.
The radius of each marshmallow is half of the diameter, so it would be 0.635/2 cm. Using that, we can calculate the volume of one marshmallow.
However, before we do the math, I have to say I find it hilarious that we're about to calculate the height of a marshmallow mountain! Okay, back to business.
So, the volume of each marshmallow is V = π(0.3175)^2 * 2.54 cm^3. Plugging this into the formula, we get:
V = 2.5301 cm^3.
Now, to find the total volume of all the marshmallows, we can multiply the volume of one marshmallow by the number of marshmallows:
Total Volume = (2.5301 cm^3) * (6.022 x 10^23).
Now comes the part where we convert things to kilometers. We know that 1 cm^3 is equal to 1 x 10^-6 km^3. So, to convert our total volume to kilometers, we can multiply it by the conversion factor:
Total Volume (in km^3) = (2.5301 cm^3) * (6.022 x 10^23) * (1 x 10^-6 km^3/cm^3).
Now, hold on to your marshmallow hat, because things are about to get wild! When you do the calculations, you'll find that your mini marshmallows would extend approximately 39,380,000,000 kilometers above the surface! That's a whole lot of marshmallows, my friend!
I hope this answer helped you, and perhaps brought a smile to your face along the way!
To find the height above the surface that the mini marshmallows will extend, we need to calculate the volume they occupy and convert it to kilometers.
First, let's calculate the volume of one mini marshmallow. The mini marshmallow can be approximated as a cylinder, so we can use the formula for the volume of a cylinder:
V = π * (r^2) * h
where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the radius of the cylinder, and h is the height of the cylinder.
Given that the diameter of the mini marshmallow is 0.635 cm, the radius (r) would be half of that:
r = 0.635 cm / 2 = 0.3175 cm
Converting the radius to meters (1 cm = 0.01 m):
r = 0.3175 cm * 0.01 m/cm = 0.003175 m
The height (h) of the mini marshmallow is given as 2.54 cm, which we'll convert to meters:
h = 2.54 cm * 0.01 m/cm = 0.0254 m
Now, let's calculate the volume of one mini marshmallow:
V = π * (0.003175 m)^2 * 0.0254 m = 2.5478435 × 10^(-8) m^3
To find out how many mini marshmallows are needed to cover the land area of California, we divide the land area by the area of one mini marshmallow:
Number of mini marshmallows = Land area of California / Area of one mini marshmallow
The area of one mini marshmallow can be calculated using the formula for the area of a circle:
A = π * (r^2)
Given that the diameter of the mini marshmallow is 0.635 cm, the area (A) can be calculated as:
A = π * (0.003175 m)^2 = 3.17627377 × 10^(-6) m^2
Now, let's calculate the number of mini marshmallows needed to cover the land area of California:
Number of mini marshmallows = (1.560 × 10^5 mi^2) / (3.17627377 × 10^(-6) m^2)
To compare the units, we'll convert the land area of California from square miles to square meters:
1 square mile = 1.609344^2 square kilometers
1 square mile = 2.58998811 × 10^6 square meters
Land area of California = 1.560 × 10^5 mi^2 * (2.58998811 × 10^6 m^2/mi^2)
Land area of California = (4.042982541 × 10^11) m^2
Number of mini marshmallows = (4.042982541 × 10^11 m^2) / (3.17627377 × 10^(-6) m^2)
Number of mini marshmallows = 1.27323764682 × 10^17
Finally, let's calculate the height above the surface in kilometers:
Height = (Total volume of mini marshmallows) / (Land area of California)
Total volume of mini marshmallows = V * (Number of mini marshmallows)
Height = (2.5478435 × 10^(-8) m^3) * (1.27323764682 × 10^17) / (4.042982541 × 10^11 m^2)
Height = 8.04553810012 kilometers
Therefore, the mini marshmallows would extend approximately 8.045 kilometers above the surface of California.