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Question 1025738: A Modern Design magazine is having a contest to design the best ice cube in one of three shape. Find and simplify the volume-to-surface-area ratio for each of the three possible ice cube shapes. Also, choose what you think is the best size for that shaped ice cube. a CUBE with an S.A. = 6s^2 and V = s^3
a SPHERE with an S.A. = 4pi r^2 and V = 4/3 pi r^3 or a CYLINDER with an S.A. = 2pi r^2 + 2pi rh V = pi r^2h.
Use the following guidelines to make your choice and complete your contest entry.
Use the ratios to choose the best shape for an ice cube--a cube, sphere or cylinder. Consider these questions as you make your decision.
A. Since the purpose of an ice cube is to keep a drink cold, is it better for an ice cube to have a high volume or a low volume?
B. Since heat touching the surface of an ice cube causes it to melt, is it better for an ice cube to have a large surface area or a small surface
area?
C. Which volume-to-surface-area ratio would be better for an ice cube
—the lowest possible or the highest possible?
D. How does increasing the size of an object affect its volume-to-surface-area ratio?
I choose to use the Cube shape for the Ice Design. I don't understand how to do this. I email my teacher but no response. This is keeping me from going to tne next assignment. PLEASE HELP!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! using the same volume of water to make the ice cube, then the ice cube with the smallest volume to surface area ratio will have the largest surface area.
in general, the smaller the surface area, the longer it takes for the ice cube to melt.
your choice of ice cube is therefore a matter of choice.
do you want the ice cube to melt slower or do you want the ice cube to melt faster?
the answer is not absolute because different drinks require different icing speeds.
some cocktails are mixed in cracked ice.
this makes the fastest cooling rate.
then the drink is poured without the ice in the drink.
you have a cool drink with minimum dilution of the alcohol in the drink.
other drinks, like a shot of courbon or whiskey desire a slower melting rate to only a small amount of melted ice is dissolved in the drink.
in other words, it's purely subjective and it is dependent on the type of drink that is consumed.
another problem is with cylindrical shaped ice.
the height can vary.
i'll assume that the height of the cylindrical ice cube is the same length as the side of the square ice cube for comparison purposes.
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first we'll look at the cube shaped ice cube.
volume = s^3 and surface area = 6s^2.
volume to surface ratio = s^3 / 6s^2 = s/6.
we will assume that each ice cube has a volume of 16 cubic centimeters.
that's approximately 1 cubic inch give or take a small percentage.
the volume is therefore 16 cubic centimeters.
solve for the length of the side to get s = cube root of 16 = 2.519842 truncated to 6 decimal places.
every measure will be truncated to 6 decimal places. this gives you plenty of detail without getting ridiculous about it.
the surface area is 6 * s^2 = 6 * 2.519842^2 = 38.097625
the volume to surface area is 16 / 38.097625 = .419973
note that s/6 = 2.519842 / 6 = .419973
this confirms the general formula of volume to surface area ratio is correct.
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now we'll look at the sphere.
the volume of a sphere is equal to 4/3 * pi * r^3.
the surface area of a sphere is equal to 4 * pi * r^2.
the volume to surface area is therefore equal to 4/3 * pi * r^3 / (4 * pi * r^2).
this simplifies to 4/3 * r / 4, which further simplifies to r/3.
r is the radius of the sphere.
we'll take a sphere with the same volume of 16.
4/3 * pi * r^3 = 16
divide both sides of this equation by pi and multiply both sides of this equation by 3/4 to get:
r^3 = 16 / pi * 3/4.
take the cube root of both sides of this equation to get:
r = (16/pi*3/4)^(1/3) = 1.563185
the radius is equal to 1.563185.
replace r in the original equation and you will find that the volume = 16.
this conmfirms the calculation of the length of the radius is correct.
the surface area of the sphere is equal to 4 * pi * r^2.
when r = 1.563185, the surface area is equal to 30.706532.
the volume to surface area is therefore 16 / 30.706532 = .521061.
note that r/3 = 1.562185 / 3 = .521061, so the formula for the volume to surface area is correct.
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now we'll look at the cylindrical shaped ice cube.
we are assuming the the height of the cylinder is the same as the length of the side of the cube shaped ice cube.
that means the height of the cylindrical shaped ice cube is equal to 2.519842.
the volume is 16.
we'll solve for the radius as follows:
pi * r^2 * h = 16
divide both sides of this equation by pi * h to get:
r^2 = 16 / (pi * h)
take the square root of both sides of this equation to get:
r = square root of (16 / (pi * h)).
since h = 2.519842, we get:
r = square root of (16 / (pi * 2.519842) = 1.421668
we have:
height = 2.519842
radius = 1.421668.
the surface area of the cylindrical shaped ice cube uses the formula of:
surface area = 2 * pi * r^2 + 2 * pi * r * h.
when h = 2.519842 and r = 1.421668, the surface area becomes:
surface area = 2 * pi * (1.421668)^2 + 2 * pi * 1.421668 * 2.519842.
resolve this formula to get surface area = 35.207969.
the volume to surface area is therefore equal to 16 / 35.207969 = .454442.
the formula for the volume to surface area ratio is derived as follows.
volume = pi * r^2 * h.
surface area = 2 * pi * r^2 + 2 * pi * r * h
the volume to surface area is therefore (pi * r^2 * h) / (2 * pi * r^2 + 2 * pi * r * h).
factor out 2 * pi * r from the denominator and you get:
volume to surface area ratio = (pi * r^2 * h) / (2 * pi * r) * (r + h).
simplify to get:
volume to surface area ratio = (r * h) / (2 * (r + h)).
height = 2.519842
radius = 1.421668.
when r = 1.421668 and h = 2.519842, this ratio becomes:
1.421668 * 2.519842) / (2 * (1.42168+ 2.519842)) = .454442.
since this is the same ratio as calculated earlier, the formula for volume to surface area is correct.
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you have the following statistics from the three shapes.
cube shaped sphere shaped cylinder shaped
volume 16 16 16
surface area 38.097625 30.706532 35.207969
v/sa ratio .419973 .521061 .454442
sa/v ratio 2.381101 1.19158 2.200498
v = volume
sa = surface area
the cube shaped has the largest surface area therefore it will cool the fastest.
the sphere shaped has the smallest surface area therefore it will cool the slowest.
the cylinder shaped is somewhere in between therefore it's a compromise between the sphere shaped and the cube shaped.
choose your preference for cooling rate and then that's your pick.
believe me, this is a lot more complicated than it looks.
a search on the web shows all kinds of relationship with different situations.
there is no one perfect shape.
it's all a matter of preference and, possible, prejudice.
as a matter of practically, the cube shaped is the easiest to form.
the sphere shaped and the cylindrical shaped are harder to form.
your last question had to do with size and its impact on the volume to surface area ratio.
with the cube, the formula for v/sa is s/6.
as s increases, the ratio will get higher, because the denominator remains the same.
this means that you will have less surface area relative to the volume.
with the sphere, the formula for v/sa is r/3.
as the radius gets larger, the ratio will also get larger.
this means that you will have less surface area relative to the volume.
with the cylinder, the formula for v/sa is rh/(2*(r+h)).
this is trickier to just observe to see which way i goes.
my guess is it gets larger (same as the others).
my test.
assume r and h are equal to 1.
the ratio is 1*1/(2*(1+1)) = 1/(2*2) = 1/4.
assume r and h are equal to 2.
the ratio is 2*2/(2*(2+4) = 4/(2*6)) = 4/12 = 1/3.
1/3 is larger than 1/4 (.333 versus .25), therefore, as the size gets larger, the ratio gets larger as well.
it appears that, in all of these, the volume to surface area ratio gets larger as the size gets larger.
this means that the surface area to volume ratio gets smaller as the size gets larger.
surface area to volume ratio is the reciprocal of volume to surface area ratio.
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