Question 167479: Hello, i was given a problem by my teacher and did not understand it:
When 0.5 cm was planed off of each of the 6 faces of a wooden cube its volume decreased by 169 cm^2. Find its new volume.
Thank You.
Found 2 solutions by nerdybill, gonzo:
Answer by nerdybill(7384)   (Show Source):
You can put this solution on YOUR website! When 0.5 cm was planed off of each of the 6 faces of a wooden cube its voume decreased by 169 cm^2. Find its new volume.
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A "cube" implies that all edges are the same length.
Volume of a cube is "length of an edge" cubed.
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Let x = original length of one edge
then
x-1 = edge of new cube
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x^3 - 169 = (x-1)^3
x^3 - 169 = (x-1)(x-1)(x-1)
x^3 - 169 = (x^2-2x+1)(x-1)
x^3 - 169 = (x)(x^2-2x+1)-(x^2-2x+1)
x^3 - 169 = (x^3-2x^2+x)-(x^2-2x+1)
x^3 - 169 = x^3-2x^2+x-x^2+2x-1
x^3 - 169 = x^3-3x^2+x+2x-1
x^3 - 169 = x^3-3x^2+3x-1
-169 = -3x^2+3x-1
0 = -3x^2+3x+168
dividing both sides by 3:
0 = -x^2+x+56
0 = x^2-x-56
0 = (x-8)(x+7)
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x = {-7, 8}
We can toss out the negative solution so we end up with:
x = 8 cm (length of an edge of the original cube)
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Length of new cube edge:
x-1 = 8-1 = 7 cm
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Finally, volume of new cube is:
7^3 = 343 cubic cm
Answer by gonzo(654)  (Show Source):
You can put this solution on YOUR website! here's what i think.
the volume of the cube = s^3 (side * side * side)
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if you plane off .5 centimeters from each face, then each side will have 1 centimeter taken off of it (.5 centimeters from each end).
i will explain this better later.
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the volume of the planed cube will therefore be (s-1)^3.
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the volume of the planed cube is 169 cubic centimeters less than the volume of the unplaned cube.
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what this says is that s^3 - (s-1)^3 = 169 cm^3.
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i worked the equation which i will detail later.
for now, i will tell you that the side of the unplaned cube is 8 centimeters.
this makes the side of the planed cube 7 centimeters.
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(8cm)^3 = 512 cubic centimeters
(7cm)^3 = 343 cubic centimeters
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512 cm^3 - 343 cm^3 = 169 cm^3.
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this satisfies the original statements of the problem so i assume that the values for each side of the planed cube and the unplaned cube are correct.
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now some details.
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first the equation:
s^3 - (s-1)^3 = 169 is where i started from.
multiplying out, this equation becomes:
s^3 - (s^3 - 3s^2 + 3s - 1) = 169
remove the parentheses:
s^3 - s^3 + 3s^2 + 3s - 1 = 169
combine like terms:
3s^2 + 3s - 1 = 169
subtract 169 from both sides of the equation and combine like terms:
3s^2 + 3s - 168 = 0
divide both sides of equation by 3:
s^2 + s - 56 = 0
factor:
(s-8)*(s+7) = 0
roots are:
s = 8
or
s = -7
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since a negative measurement is not possible, the answer has to be s = 8.
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each side measures 8 centimeters in length.
this is equivalent to:
length = 8, width = 8, height = 8 since a cube is a rectangular solid where length = width = height.
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the assumption of 1 centimeter off of each dimension was made by visualizing what happens to the cube when you remove .5 centimeters off of each face.
look at it from one side and you can see that removing .5 centimeters from each end reduces that dimension by 1 centimeter (.5 from each end).
look at it from all 3 sides and you can then determine that each dimension is reduced by that same amount.
if you have trouble visualizing this, then take some soap or styro-foam or anything easy to cut and remove a measured amount from each face. you will see that each dimension has been reduced by double that number.
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hope this helps.
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