Derive a rule of thumb that applies with the runtime is proportional to the sixth root of the input size. The rule should apply when the input size is cubed. Make sure so show the math and eventually translate into a sentence of the form "When we cube the input size...". Do not use $"$n$"$ in your sentence. Say "the original input size".

$2)$ Derive a rule of thumb that applies with the runtime is proportional to sqrt$($n$)($log$($n$))2,$ where n is the input size. The rule should apply when the input size is cubed. Make sure so show the math and eventually translate into a sentence of the form "When we cube the input size...". Do not use $"$n$"$ in your sentence. Say "the original input size".

$3)$ The runtime of a given computer program is proportional to $5$n$,$ where n is the input size. Suppose the program takes $16$ms to process input of size $20.$ How long will it take to process input of size $23?$ Use a rule of thumb and show all calculations.

$4)$ Suppose a program takes time proportional to $($log$($n$))2,$ where n is the input size. If the program takes $30$ ms to process input of size $100,$ how long will it take to process...

a$)$ input if size $10,000?$

b$)$ input of size $100,000,000?$

Justify your answers using on or more rules of thumb. $($Do not solve an equation to get the answer.$)$