Question 1027670: If the surface of a cube is 6x^2 -36x +54, what is the expression for the volume of the cube?
Found 2 solutions by mananth, Theo:
Answer by mananth(16946)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! formula for surface area of a cube is 6s^2.
s represents the length of the side of the cube.
that means that 6s^2 = 6x^2 - 36x + 54.
divide both sides of this equation by 6 and you get:
s^2 = x^2 - 6x + 9
factor that quadratic equation on the right side of the equation and you get:
s^2 = (x-3) * (x-3) = (x-3)^2
take the square root of both sides of the equation to get:
s = plus or minus (x-3).
since s can't be negative, then s = (x-3).
note that the value of x would have to be greater than 3, otherwise the length of the side would be negative.
the formula for the volume of a cube is s^3.
that means the volume of the cube must be equal to (x-3)^3.
since (x-3)^2 = x^2 - 6x + 9, then (x-3)^3 must be equal to:
(x-3) * (x^2 - 6x + 9) which would then be equal to x^3 - 9x^2 + 27x - 27.
how do you confirm this is correct?
assume the side of the cube is equal to 5 units.
the surface area would be 6s^2 = 6 * 5^2 = 6 * 25 = 150 square units.
the volume would be s^3 = 5^3 = 125 cubic units.
we had previously shown that s = x-3.
this means that x = s + 3.
if s = 5, this means that x = 8.
our formula for surface area is 6 * s^2 = 6.
since s = (x-3), then the surface area = 6 * (x-3)^2 = 6x^2 - 36x + 54.
replace x with 8 in these formulas and you will see that 6 * (x-3)^2 = 6 * 5^2 = 6 * 25 = 150 square units.
you will also see that 6x^2 - 36x + 54 = 6 * 8^2 - 36 * 8 + 54 = 384 - 288 + 54 = 150 square units.
the volume of the cube is equal to s^3.
when s = 5, this gets you the volume of the cube = 125 cubic units.
when s = 5, x = 8.
the volume of the cube = (x-3)^3 = (8-5)^3 = 5^3 = 125 cubic units.
the volume of the cube is also equal to x^3 - 9x^2 + 27x - 27.
when x = 8, this becomes (8)^3 - 9*8^2 + 27*8 - 27 which becomes equal to 512- 576 + 216 - 27 which becomes equal to 125 cubic units.
the formulas look good.
the expression for the volume of the cube would be (x-3)^3 or x^3 - 9x^2 + 27x - 27, depending on how you want to show it.
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