Question 991842: The diameter of a sphere is reduced by 10%. By what percent will the volume be reduced?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! since diameter = 2 times the radius, then if you reduce the diameter by 10%, you also reduce the radius by 10%.
for example:
assume the diameter is 10.
reduce it by 10% and the diameter is now 9.
since the diameter is 10, the radius has to be 10/2 = 5.
since the reduced diameter is 9, the reduced radius has to be 9/2 = 4.5.
5 - 4.5 is equal to .5 and .5 / 5 is equal to .10 which is equal to 10%, so the radius has been reduced by 10% as well.
the formula for the volume of a sphere is 4/3 * pi * r^3.
when you reduce the radius by 10%, the resulting radius is 90% of the original radius.
this is because r - .10 * r is equal to r * (1 - .10) which is equal to r * .9.
reduce the radius by 10% and the formula becomes v = 4/3 * pi * (.9*r)^3 which becomes v = 4/3 * pi * .9^3 * r^3 which then becomes v = 4/3 * pi * .729 * r^3.
this can be rewritten as v = .729 * 4/3 * pi * r^3.
the volume has been reduced by (1 - .729) = .271, or 27.1%.
let's take an example:
assume the diameter of the sphere is 24 inches.
the radius then has to be 12 inches because the length of the radius is half the length of the diameter.
the volume of the sphere is 4/3 * pi * 12^3 which is equal to 4/3 * pi * 1728.
reduce the diameter by 10%.
24 - .10 * 24 = 24 - 2.4 = 21.6
this is the same as .9 * 24 which is equal to 21.6.
this is because 24 - .10 * 24 is equal to 24 * (1 - .10) which is equal to .9 * 24.
the radius of this sphere is equal to 21.6 / 2 = 10.8
the volume of this sphere is equal to 4/3 * pi * (10.8)^3 which is equal to 4/3 * pi * 1259.712.
the volume of the original sphere is 4/3 * pi * 1728.
the volume of the reduce sphere is 4/3 * pi * 1259.712.
the difference in the volume is 4/3 * pi * 1728 - 4/3 * pi * 1259.712.
this can be rewritten as 4/3 * pi * (1728 - 1259.712) which can then be simplified to 4/3 * pi * 468.288.
the ratio of this difference to the original volume is:
(4/3 * pi * 468.288) / (4/3 * pi * 1728).
4/3 * pi / 4/3 * pi cancels out because it is equal to 1 and you are left with:
the ratio of the difference is equal to 468.288 / 1728 which is equal to .271 which is equal to 27.1%.
reduce the diameter of the sphere by 10% and you reduce the volume of the sphere by 27.1%.
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