Question : If $x+y+z=10$, $x y+y z+z x=25$ and $x y z=100$, then what is the value of $(x^3+y^3+z^3)$?
Option 1: 450
Option 2: 540
Option 3: 550
Option 4: 570
Correct Answer: 550
Solution :
Given that $x + y + z = 10$, $xy + yz + zx = 25$, and $xyz = 100$.
Now, $(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)$
$⇒100 = x^2 + y^2 + z^2 + 2\times25$
$⇒x^2 + y^2 + z^2 = 100 - 50 = 50$
Now, we can substitute $x + y + z = 10$, $x^2 + y^2 + z^2 = 50$, and $xyz = 100$ into the formula,
$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$
$⇒x^3 + y^3 + z^3 = 10\times50 - 10\times25 + 3\times100 = 500 - 250 + 300 = 550$
Hence, the correct answer is 550.
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