- #1

- 2

- 0

i)How many students will be aware of the rumour after 7 days.

ii)How long will it take for 850 students to hear the rumour

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- MHB
- Thread starter Help_me1
- Start date

In summary, after 1 day, 5 students will be aware of the rumour and 10 students will be aware of the rumour after 7 days. It will take 850 students 7 days to be aware of the rumour.f

- #1

- 2

- 0

i)How many students will be aware of the rumour after 7 days.

ii)How long will it take for 850 students to hear the rumour

- #2

- 1,103

- 1

$(1000-r)$ = number who have not heard the rumor

$t$ is in days

$\dfrac{dr}{dt} = k \cdot r(1000-r)$, where $k$ is the constant of proportionality

you are given $r(0)=5$ and $r(1) = 10$

see what you can do from here …

- #3

- 2

- 0

Thank you

- #4

- 921

- 2

dr/dt= kr(1000- r)

dr/[r(1000-r)]= kdt

To integrate on the left, separate using "partial fractions"- find constant A and B such that 1/[r(1000- r)]= A/r+ B/(1000- r)

Multiplying on both sides by r(1000- r)

1= A(1000- r)+ Br

Let r= 0: 1= 1000A so A= 1/1000.

Let r= 1000: 1= 1000B so B= 1/1000

1000 dr/r+ 1000 dr/(1000- r)= kdt

The integral of 1000 dr/r is 1000 ln(|r|).

To integrate 1000 dr/(1000- r) let u= 1000- r so that du= -dr.

then dr/(1000- r)= -du/u.

The integral is -1000 ln(|u|)= -1000 ln(|1000- r|)

We have 1000 ln(|r|)-1000 ln(|1000- r|)= kt+ C.

$1000 ln\left(|\frac{r}{1000- r}|\right)= kt+ C$

$ln\left(\left(\frac{r}{1000- r}\right)^{1000}\right)= kt+ C$

(I have dropped the absolute value since this is to an even power.)

Taking the exponential of both sides

$\left(\frac{r}{1000- r}\right)^{1000}= C' e^{kt}$

where $C'= e^C$.

Now, we are given that r(0)= 5 and r(1)= 10.

Setting t= 0, r= 5

$\frac{5}{9995}= \frac{1}{1999}=C'$

Setting t= 1, r= 10

$\frac{10}{9990}= \frac{1}{999}= \frac{e^k}{1999}$

$e^k= \frac{1999}{999}$

so $k= ln\left(\frac{1999}{999}\right)$

and $e^{kt}= (e^k)^t= \left(\frac{1999}{999}\right)^t$

$r(t)= \frac{\left(\frac{1999}{999}\right)^t}{1999}$.

- #5

- 1

- 0

You are putting A and B values in a wrong way ...these values will come in fraction

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