For sorting problem, there are two categories of algorithms: one sorts by reducing the problem size by one at each step, and the other category of algorithms divides the problem into two smaller problems, solves each smaller problem recursively and then combines the two sorted lists. The first category of algorithms are not optimal and time complexity of these algorithms are $O(n^2).$ Whereas, some of the algorithms in the second category have theoretically optimal time complexity. But theoretically optimal algorithms are not always fastest. In many cases, if input size is greater than a constant then only algorithm is faster. In this homework problem you have to find the constant below which the a non$-$optimal algorithm is faster and use it to answer the questions. Suppose your are comparing implementations of insertion sortand heap sort algorithms on the same machine. For input size $n,$ insertion sort runs in $10n^2$ steps, while the heap sort runs in $20480nlo{g}_{2}n$ steps.

$($a$)$ For which values of $n$ bubble sort beats the heap sort? Note that $n1.$

$($b$)$ Explain how you found your answer and show your work.