Assume that a mergesort algorithm in the worst case takes 30 seconds for an input of size 64. Which of the following most closely approximates the maximum input size of a problem that can be solved in 6 minutes

The time complexity of the merge sort algorithm is O(n log n). This means that the time it takes to sort an array increases linearly with the size of the array and logarithmically with the size of the array.

Given that the algorithm takes 30 seconds to sort an array of size 64, we can set up the following equation to find the constant of proportionality, k:

30 = k * 64 * log2(64)

Solving for k gives us:

k = 30 / (64 * log2(64)) = 30 / (64 * 6) = 0.078125

Now, we want to find the maximum input size that can be sorted in 6 minutes, or 360 seconds. We can set up the following equation and solve for n:

360 = 0.078125 * n * log2(n)

This is a difficult equation to solve directly, but we can use a method such as binary search or a numerical method to find the value of n that satisfies this equation.

However, since the question asks for an approximation, we can make an educated guess. Given that the time complexity is O(n log n), doubling the size of the input will more than double the time it takes to sort. Therefore, we can expect that the maximum input size that can be sorted in 6 minutes is significantly less than 64 * (360 / 30) = 768.

So, the maximum input size that can be sorted in 6 minutes is likely to be in the hundreds, but less than 768.