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------------------------------------------------ An algorithm takes 30 seconds for an input of size 1000. If the algorithm is quadratic, how large a problem can be solved in two minutes?

A. 2000
B. 4000
C. 6000
D. 60000
E. None of the above

To find the largest problem that can be solved in two minutes, we set up a proportion using the given information about the algorithm's time complexity. Solving the proportion gives us the answer as 2000. Option A is correct.

Explanation:

To determine how large a problem can be solved in two minutes, we need to find the number of seconds it takes for the algorithm to solve the problem of that size. We are given that the algorithm takes 30 seconds for an input of size 1000, and since the algorithm is quadratic, we can assume that the time it takes to solve a problem is proportional to the square of the input size. We can set up a proportion to find the size of problem that can be solved in two minutes:

(1000^2) / 30 = (x^2) / 120 From this proportion, we can solve for x, which represents the size of the problem. Simplifying the equation gives us: x = sqrt((1000^2) * 120 / 30) = 2000. Therefore, the largest problem that can be solved in two minutes is 2000. So, the correct answer is option a) 2000.