use Rouths stability criterion to determine the range of K for s^3 + 20s^2+ 100s + K10 =0

To apply Routh's stability criterion, we need to first write the characteristic equation in standard form:

s^3 + 20s^2 + 100s + K/10 = 0

Dividing both sides by the coefficient of s^3, we get:

s^3 + 20/1 s^2 + 100/1 s + K/10 = 0

Now, we can construct the Routh array:

| s^3 | 1 | 100/1 | | s^2 | 20/1 | K/10 | | s^1 | (100 - 20K/100)/1 | 0 | | s^0 | K/10 | 0 |

The first column of the Routh array should have all positive entries to ensure stability. Thus, we set the first entry to 1 and find the condition for the second entry to be positive:

20/1 > 0

This condition is always satisfied, so we can move on to the next row. We find the condition for the third entry to be positive:

(100 - 20K/100)/1 > 0

Simplifying, we get:

K < 500

Finally, we find the condition for the fourth entry to be positive:

K/10 > 0

This condition is always satisfied.

Therefore, the range of K for stability is:

0 < K < 50