Final answer:
To maximize the product of the lengths a and b of an 18 inch line, where their product is 18, they should be equal. The lengths a and b should both be 9 inches, therefore the answer is (a) a=b.
Step-by-step explanation:
The question asks how to divide an 18 inch line into two parts of lengths a and b such that the product a × b is equal to 18 and is maximized. To maximize the product of two numbers that add up to a constant, the numbers should be as close to each other as possible. According to the arithmetic mean-geometric mean inequality, the arithmetic mean is always greater than or equal to the geometric mean, with equality holding when the two numbers are equal.
Therefore, for two positive numbers a and b where a + b is constant, the product a × b is maximized when a is equal to b. In this case, we can divide the 18 inch line into two parts of 9 inches each, so a = b = 9 and a × b = 18. Hence, when a is equal to b, the product is maximized and the answer is (a) a = b.