Question : The cube of the sum of two given numbers is 1728, while the product of the two given numbers is 32. Find the positive difference between the cubes of the two given numbers.
Option 1: 448
Option 2: 576
Option 3: 480
Option 4: 512
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Correct Answer: 448
Solution : Let the two numbers be $a$ and $b$ $(a+b)^{3}=1728$ ⇒ $a+b=\sqrt[3]{1728}=12$............I $ab=32$ given, Now, $(a+b)^2-(a-b)^2=4ab$ ⇒ $(a-b)^2 = 12^2-4\times 32$ ⇒ $(a-b)^2 = 16$ ⇒ $a-b=4$.................II By adding eq. I and eq. II $2a=16$ ⇒ $a=8$ Put $a=8$ in eq. I ⇒ $8 +b =12$ ⇒ $b=4$ So from this, we know the numbers are $8$ and $4$. The positive difference between the cubes of these numbers $=(8^{3}-4^{3})=512-64=448$ Hence, the correct answer is 448.
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