When 0.5 cm was planed off each of the six faces of a wooden cube, its volume decreased by 169cm^3. Find its new volume.

Guest Sep 26, 2014

#1**+11 **

Let's call the original volume V_{o} ......... and let's call the new volume V_{n}

And V_{o} = s^{3 } = V_{n} + 169 where s is the original side length

Note that, if .5 cm is shaved off each face, then each side is now just (s-1)cm long......so we have...

V_{n} = (s - 1)^{3 } = s^{3} - 3s^{2} + 3s - 1 and substituting V_{n} + 169 for s^{3} we have

V_{n} = [ V_{n} + 169] - 3s^{2} + 3s - 1 and subtracting V_{n} from both sides and rearranging, we have

3s^{2} - 3s - 168 = 0 divide both sides by 3

s^{2} - s - 56 = 0 factor this

(s - 8)(s +7) = 0 so, s = 8 or s = -7 (reject the negative answer)

So, s = 8 and that's the original side length

So the new side length = (s - 1) = (8 - 1 ) = 7 ..... and the new volume is 7^{3} = 343cm^{3}

Note that the original volume = 8^{3} = 512cm^{3}

And 512cm^{3} - 343cm^{3} = 169cm^{3} .....and that's the difference between the old volume and the new volume !!!

CPhill
Sep 26, 2014

#1**+11 **

Best Answer

Let's call the original volume V_{o} ......... and let's call the new volume V_{n}

And V_{o} = s^{3 } = V_{n} + 169 where s is the original side length

Note that, if .5 cm is shaved off each face, then each side is now just (s-1)cm long......so we have...

V_{n} = (s - 1)^{3 } = s^{3} - 3s^{2} + 3s - 1 and substituting V_{n} + 169 for s^{3} we have

V_{n} = [ V_{n} + 169] - 3s^{2} + 3s - 1 and subtracting V_{n} from both sides and rearranging, we have

3s^{2} - 3s - 168 = 0 divide both sides by 3

s^{2} - s - 56 = 0 factor this

(s - 8)(s +7) = 0 so, s = 8 or s = -7 (reject the negative answer)

So, s = 8 and that's the original side length

So the new side length = (s - 1) = (8 - 1 ) = 7 ..... and the new volume is 7^{3} = 343cm^{3}

Note that the original volume = 8^{3} = 512cm^{3}

And 512cm^{3} - 343cm^{3} = 169cm^{3} .....and that's the difference between the old volume and the new volume !!!

CPhill
Sep 26, 2014