Question 228545: When 0.5 cm was planed off each of the six faces of a wooden cube, its volume decreased by 169 cm^3. Find its new volume.
Answer by rapaljer(4671)   (Show Source):
You can put this solution on YOUR website! Let x = new side of cube (after planing!)
x+.5 = original side of cube
Original volume = (x+.5)^3
New volume = x^3
Original volume - New volume = 169
(x+.5)^3-x^3=169
(x+.5)^3 - x^3-169=0
Because this appears to be a difficult algebra equation to solve, I can solve this using a graphing calculator. Graph the function
y= (x+.5)^3 - x^3 -169
and find the x intercept(s). These value(s) of x will be the solution(s) to the equation. I'm using a TI-83 or TI-84, and I know the value of x must be postitive or else there is no cube. I'm guessing the value of x will be less than 20, so set the window of the calculator for x and y to be between 0 and 20.
Notice that the graph crosses the x axis at between x=10 and x=11. It turns out to be approximately 10.36. Unfortunately, this method does not give me the exact value. Believe me, there IS an exact value, but it's just too much work to find it! Actually, the calculator gives a fairly accurate solution of x=10.36347414. Sometimes the calculator will actually convert this number to a fraction and give the exact value, but in this case it does not. This leads me to believe that the actual value comes out in a radical form.
Now, the question was to find the NEW volume, which would be
V=x^3
V= 10.36347414^3
V=1113.053666 cm^3 approximately
Check: The ORIGINAL volume is
V=(x+.5)^3
V=10.86347414^3
V=1282.053665 cm^3
Notice that the difference in the volumes is almost exactly 169. If anyone wants more details about the use of the graphing calculator, send me an Email at rapaljer@scc-fl.edu. Or if you REALLY want the exact value!!
Thanks for a very neat problem.
Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus
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