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

Answer:

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.

Step-by-step 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:

$Half town= \frac{800}{2}=400$ 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:

$400=120e^4^k$

$\frac{400}{120} =e^4^k$

$\frac{10}{3}=e^4^k$

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

$Ln(\frac{10}{3} )= 4*k\\k=\frac{Ln(\frac{10}{3} )}{4}$

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

$k=\frac{Ln\frac{10}{3}}{4}$

When we replace in equation 1 we have:

$720=120*e^\frac{Ln\frac{10}{3} }{4} ^*^t$

$\frac{720}{120}=e^\frac{Ln\frac{10}{3}}{4}^t\\6=e^\frac{Ln\frac{10}{3}}{4}^t$

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

$Ln 6 = \frac{Ln\frac{10}{3}}{4}*t$  

$4*Ln 6 = Ln\frac{10}{3}*t\\t=\frac{4*Ln 6 }{Ln\frac{10}{3}}$

$t=\frac{4*1.79}{1.20} \\t=\frac{7.16}{1.20} \\t=5.9 hours$

We can find how many minutes are 0.9 hours:

$0.9hours\frac{60 minutes}{1hour} = 54 minutes$

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.