Question

A small town has 800 inhabitants. At 8 AM, 120 people have heard a rumor. By noon half the town has heard it. At what time will 90% of the population have heard the rumor? (Do not round k in your calculation. Round the final answer to one decimal place.)

Answer 1

At what time will 90% of the population have heard the rumor?

The rumor was heard by 90% of the population in 5.9 hours or 5 hours and 54 minutes

So, at 1:54 pm, 90% of the population had heard the rumor.

Explanation:

To resolve this exercise we need to know the exponential model:

`P_(_t_)= P_0 *e^k^t` (1)

Where:

`P_(_t_):` the quantity inhabitants in certain time who heard the rumor

`P_0:` Initial people who heard the rumor

k: constant

t: time frame

We know in 4 hours (
`12-8=4` hours) half the town has heard the rumor because:

`Half town= (800)/(2)=400` inhabitants

With this information we can find the constant (k), because we have all the information in
`t=4 hours`

`P_(_4_)=` 400 people

`P_0=` 120 people

t= 4 hours

When we replace in equation 1 we have:

`400=120e^4^k`

`(400)/(120) =e^4^k`

`(10)/(3)=e^4^k`

We multiply by natural logarithm on both sides of this equation and we have:

`Ln((10)/(3) )= 4*k\\k=(Ln((10)/(3) ))/(4)`

With the constant (k) we can find at what time 90% of the population have heard the rumor

`800*0.9=720 people` (90% of the population)

So we have:

`P_(_t_)=` 720 people

`P_0=` 120 people

`k=(Ln(10)/(3))/(4)`

When we replace in equation 1 we have:

`720=120*e^(Ln(10)/(3) )/(4) ^*^t`

`(720)/(120)=e^(Ln(10)/(3))/(4)^t\\6=e^(Ln(10)/(3))/(4)^t`

We multiply by natural logarithm on both sides of this equation and we have:

`Ln 6 = (Ln(10)/(3))/(4)*t\\`

`4*Ln 6 = Ln(10)/(3)*t\\t=(4*Ln 6 )/(Ln(10)/(3))`

`t=(4*1.79)/(1.20) \\t=(7.16)/(1.20) \\t=5.9 hours`

We can find how many minutes are 0.9 hours:

`0.9hours(60 minutes)/(1hour) = 54 minutes`

t= 5 hours and 54 minutes

Now, we know the rumor was heard by 90% of the population in 5.9 hours or 5 hours and 54 minutes

So, at 1:54 pm, 90% of the population had heard the rumor.