Question

The rate of increase in the number,n, of people infected by a virus is modeled as being proportional to the square root of a number of people already infected. nine people were infected 5 days after the first person was infected. Form and solve a differential equation to represent this information

Answer 1

hi!

Answer 2

Okay... so rate of increase in n can be written as \(\cfrac{d n}{d t}\)

Answer 3

ok

Answer 4

And it is given that the rate is directly proportional to people already inflected i.e. n...

So, our equation becomes:

\(\cfrac{dn}{dt} = kn\)

Where, k = constant of proportionality

Answer 5

Any doubt?

Answer 6

Wait.... that should be sqrt(n)

Answer 7

Our DE will be:

\(\cfrac{dn}{dt} = k\sqrt{n}\)

Answer 8

Now can you solve this?

Answer 9

Simplify it to this:

\(\cfrac{dn}{\sqrt{n}} = kdt\)

Then integrate it...and use the condition given in the problem

Answer 10

do we integrate both sides of the equation

Answer 11

Yeah

Answer 12

ok should i be getting ln| sqrt. n |=x+C

Answer 13

You wouldn't get an ln term or x ...

Answer 14

see...

\[\int{\cfrac{dn}{\sqrt{n}} = \cfrac{\sqrt{n}}{\frac{1}{2}}} = 2\sqrt{n}\]

Answer 15

ok

Answer 16

Similarly... \(\int{k dt} = kt\)

Answer 17

o i see i was mixing my variable...

Answer 18

:)

Answer 19

thank you

Answer 20

but do we have to do anything with the part that say .. nine person was infected 5 days after the first person was infected

Answer 21

or does that part goes for part b which ask. how many days, to the nearest day, does the model predicts that it will take for 100 people to be infected ?

Answer 22

@vishweshshrimali5