Final answer:
The ratio of the squares of the given numbers is 36:144:324.
Step-by-step explanation:
The given ratios are 1:2:3. Let's assume the numbers represented by these ratios are x, 2x, and 3x.
Half the sum of these numbers is 18, so (x + 2x + 3x)/2 = 18.
By solving this equation, we get 6x/2 = 18.
Simplifying further, we have 3x = 18.
Dividing both sides by 3, we find that x = 6.
Therefore, the numbers are 6, 12, and 18.
The ratio of the squares of these numbers is (6^2):(12^2):(18^2), which simplifies to 36:144:324.
So, the correct answer is option (c) 36:144:324.