SOLUTION: A computer is infected with the Sasser virus. Assume that it infects 20 other computers within 5 minutes; and that these PCs and servers each infect 20 more machines within anothe

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: A computer is infected with the Sasser virus. Assume that it infects 20 other computers within 5 minutes; and that these PCs and servers each infect 20 more machines within anothe      Log On


   

Question 70515: A computer is infected with the Sasser virus. Assume that it infects 20 other computers within 5 minutes; and that these PCs and servers each infect 20 more machines within another five minutes, etc. How long until 100 million computers are infected?
Help with this, please? Need to show steps and I'm lost completely! Thanks a million!
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
First you have to have a good idea of what's
going on.
You've got 1 infected computer to start with
Then you've got 1 + 20 total
Each of the 20 newly infected computers infect 20 more
Now you have 1 + 20 + 20*20 infected
Each of the 20*20 newly infected computers infect 20 more
Now you have 1 + 20 + 20*20 + 20*20*20 infected
or 1+%2B+20%5E1+%2B+20%5E2+%2B+20%5E3
call this sum S and carry it out to n places
S+=+1+%2B+20%5E1+%2B+20%5E2+%2B+20%5E3+ ... ++20%5En
subtract 1 from both sides
S+-+1+=+20%5E1+%2B+20%5E2+%2B+20%5E3+ ... ++20%5En
divide both sides by 20
%28S+-+1%29+%2F+20+=+1+%2B+20%5E1+%2B+20%5E2+%2B+20%5E3+ ... ++20%5E%28n-1%29
The right side of this equation is S+-+20%5En
%28s+-+1%29+%2F+20+=+S+-+20%5En
multiply both sides by 20
S+-+1+=+20S+-+20%5E%28n%2B1%29
19S+=+20%5E%28n%2B1%29+-+1
S+=+%2820%5E%28n%2B1%29+-+1%29+%2F+19
For the case n = 4
S+=+1+%2B+20%5E1+%2B+20%5E2+%2B+20%5E3+%2B+20%5E4 or
1+%2B+20+%2B+400+%2B+8000+%2B+160000+=+168421
Now check the formula
S+=+%2820%5E%28n%2B1%29+-+1%29+%2F+19
S+=+%2820%5E5+-+1%29+%2F+19
S+=+%283200000+-+1%29+%2F+19
S+=+3199999+%2F+19
S+=+168421
The formula works, Now what does n have to be for
S to be at least 100,000,000?
let's try n = 10
S+=+%2820%5E11+-+1%29+%2F+19
S+=+%282.048%2A10%5E14+-+1%29+%2F+19
This is bigger than 10^9
Let's try n = 8
S+=+%2820%5E9+-+1%29+%2F+19
This is roughly 27%2A10%5E9
Let's try n = 7
S+=+%2820%5E8+-+1%29+%2F+19
This is roughly 1.28%2A10%5E9
So, the number of computers infected goes over
100 million between 6 and 7 5-minute intervals
To find the answer exactly,
10%5E9+=+%2820%5E%28n%2B1%29+-+1%29+%2F+19
20%5E%28n%2B1%29+-+1+=+19%2A10%5E9
20%5E%28n%2B1%29+=+19%2A10%5E9+%2B+1
%28n%2B1%29+log%2820%29+=+log%2819%29+%2B+9%2Alog%2810%29 (ignoring the +1)
1.301%2A%28n%2B1%29+=+1.279+%2B+9
1.301n+%2B+1.301+=+10.279
1.301n+=+8.978
n+=+6.901 5-minute intervals
6.901%2A5+=+34.5 minutes
So, 100 million computers are infected in
34 minutes and 30 seconds approximately