Question 159142
How many washers can be made from a cube of metal 4 inches on a side if the washers are 5/8 inch in diameter and 1/6 inch thick?The hole in the center of the washers is 1/4 inch in diameter.
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cube of metal 4 inches on a side has 4*4*4 = 64 cubic inches of metal in it.
if we let C represent the amount of metal in the cube, then
C = 64 cubic inches.
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amount of metal in the washer is the volume of the washer using the exterior dimensions minus the volume of the washer using the interior dimensions.
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washer can be represented by circular cylinder.
formula for volume of a circular cylinder is pi*r^2*h
where r is the radius and h is the height and pi is the constant 3.1415......
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exterior dimensions of the washer are 5/8 inches in diameter and 1/6 inches thick.
if diameter is 5/8 inches, then radius is 5/16 inches.
let V1 = volume of washer using exterior dimensions only.
let r = radius = 5/16
let h = height = 1/6
V1 = r^2*h*pi
V1 = (5/16)^2*(1/6)*pi
V1 = (25*1*pi)/(16*16*6)
V1 = (25*pi)/(1536)
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interior dimensions of the washer are 1/4 inches in diameter and 1/6 inches thick.
if diameter is 1/4 inches, then radius is 1/8 inches.
let V2 = volume of washer using interior dimensions only.
let r = radius = 1/8
let h = height = 1/6
V2 = r^2*h*pi
V2 = (1/8)^2*(1/6)*pi
V2 = (1*1*pi)/(8*8*6)
V2 = (pi)/(384)
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V1 represents the amount of metal in the washer if it was solid throughout (outer dimensions only).
V2 represents the amount of metal that has to be removed from the washer in order to make room for the hole (interior dimensions only)
amount of metal in the washer is then
V1 - V2 = (25*pi/1536) - (pi/384)
if we let W = amount of metal in the washer, then
W = V1-V2 = (25*pi/1536) - (pi/384), so
W = (25*pi/1536) - (pi/384)
multiplying both sides of the equation by 1536 to remove the denominator, we get
1536*W = 25*pi - 4*pi = 21*pi
dividing both sides of the equation by 1536, we get
W = 21*pi/1536
this means that each washer has 21*pi/1536 cubic inches of metal in it.
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the amount of metal available is equal to the cube of metal provided is equal to 64 cubie inches of metal.
the number of washers that can be made with this metal is determined by the amount of metal in the cube divided by the amount of metal in each washer.
let C = amount of metal in the washer.
let W = amount of metal in each washer.
let N = number of washers.
then
N = C/W
this becomes
N = 64/(21*pi)/1536)
64/(21*pi)/1536 is the same as (64*1536)/(21*pi).
equation for N becomes
N = (64*1536)/(21*pi)
N = 1490.05405 washers which is equal to 1490 washers with a small amount of metal left over.