A rumor spreads in a small town with population 20,000 people. The secret spreads throughout the town according to the function, where t represents time, in days: N(t)=20000/10+2490^.97t

A.) How many people were in the group that started the rumor?

B.) How many people have heard the rumor after 1 day? One week?

C.) At this rate, how long will it take for 1950 people to hear the rumor?

Question

A rumor spreads in a small town with population 20,000 people. The secret spreads throughout the town according to the function, where t represents time, in days: N(t)=20000/10+2490^.97t

A.) How many people were in the group that started the rumor?

B.) How many people have heard the rumor after 1 day? One week?

C.) At this rate, how long will it take for 1950 people to hear the rumor?

ria23 · Answer

$$N(t)=\frac{ 20000 }{ 10+2490e^{-0.97t} }$$

anonymous · Answer

i should make a template for this one
have seen this exact question over a dozen times in the past two years

anonymous · Answer

A.) How many people were in the group that started the rumor? 
put $t=0$  what do you get?

anonymous · Answer

hint: $e^0=1$

ria23 · Answer

It'll just be the exact same problem won't it?
10+2490?

anonymous · Answer

that is in the denomnator

anonymous · Answer

$$\frac{20,000}{10+2490}$$  compute that one

coconutjj · Answer

8 people ?

ria23 · Answer

20000/3000 which gives me a long decimal

anonymous · Answer

$$N(t)=\frac{ 20000 }{ 10+2490e^{-0.97t} }\
N(0)=\frac{ 20000 }{ 10+2490e^{0} }=\frac{20,000}{10+2490}$$

anonymous · Answer

oh no dear, what is $2490+10$ ?

ria23 · Answer

Oh... Woops... Haha
20000/2500
8 c:

anonymous · Answer

whew

anonymous · Answer

B.) How many people have heard the rumor after 1 day?
$$N(t)=\frac{ 20000 }{ 10+2490e^{-0.97t} }\
N(1)=\frac{ 20000 }{ 10+2490e^{-.097} }=\text{ use a calculator}$$

anonymous · Answer

B.) How many people have heard the rumor after 1 week 
put $t=7$ compute $$N(7)=\frac{ 20000 }{ 10+2490e^{-.097\times 7} }=\text{ use a calculator}$$

anonymous · Answer

oops i typed it in wrong in to wolf, let me redo the previous one

ria23 · Answer

The second one is almost 16...
15.714

anonymous · Answer

same answer though, not quite 9 in any case you can use the wolfram link i sent as a template if you do not want to plug this in to a calculator this would be for 7 days http://www.wolframalpha.com/input/?i=20000%2F%2810%2B2490e^%28-0.097*7%29%29

anonymous · Answer

yeah that is what i got too

ria23 · Answer

And that's B?

anonymous · Answer

yes

anonymous · Answer

C.) At this rate, how long will it take for 1950 people to hear the rumor? 
this is the only one that requires any real work 
any ideas?

ria23 · Answer

would I do the same thing as the last one, but plug in 1950 for t?

anonymous · Answer

oh no

anonymous · Answer

this one says the ANSWER is 1950, what $t$ did you plug in to get it?

anonymous · Answer

$$N(t)=\frac{ 20000 }{ 10+2490e^{-0.97t} }=1950$$ solve for $t$

ria23 · Answer

Can yhu tell me what my first step would be?

anonymous · Answer

multiply by the denominator to clear the fraction 
$$20,000=1950(10+e^{-.97t})$$

anonymous · Answer

damn typo 
$$20,000=1950(10+2490e^{-97t})$$

anonymous · Answer

then distribute teh $1950$ on the right

anonymous · Answer

or you can divide both sides by 1950 first either way

ria23 · Answer

$$20,000= 19,500+4,855,500e^{-.97t}$$

ria23 · Answer

;-;

anonymous · Answer

i believe you

anonymous · Answer

subtract 19500 from both sides

anonymous · Answer

even i can do that without a calculator 
$$500=4,855,500e^{-.97t}$$

ria23 · Answer

$$500=4,855,500e^{-.97}$$
c:

anonymous · Answer

divide by $4,855,500$

ria23 · Answer

4 million divided by 500?
Or 500 divided by the 4 million?

anonymous · Answer

second way
it is a tiny fraction 
$$\frac{1}{9771}$$

ria23 · Answer

... 9771? 
I got 9711... ;c

anonymous · Answer

now comes the only non algebra step $$\frac{1}{9771}=e^{-.97t}$$

anonymous · Answer

that's cause you did it upside down

anonymous · Answer

$$a=bx\
\frac{a}{b}=x$$

anonymous · Answer

ok now do you know what to do here?$$\frac{1}{9771}=e^{-.97t}$$

ria23 · Answer

No ;-; 
And I keep getting 9711. ;c

I flipped it around and got 1.long decimal ;C

anonymous · Answer

"no" is a fine answer 
forget the decimal

anonymous · Answer

you got $9771$ because you divided 500 by 4855500

anonymous · Answer

ack i mean that is what you did not do
you divided backwards

anonymous · Answer

it should be 
$$500\div 4855500$$ and you should get the reciprocal of $9771$ which is $\frac{1}{9771}$ a very tiny decimal, lets leave it as a fraction

ria23 · Answer

;c

ria23 · Answer

...........................

ria23 · Answer

Wait the decimal is right?

anonymous · Answer

oops typo again!

anonymous · Answer

$$\frac{1}{9711}=e^{-.97t}\
\ln(\frac{1}{9711})=-.97t$$

anonymous · Answer

and finally divide by $-.97$ to find $t$ i used this http://www.wolframalpha.com/input/?i=ln%281%2F9711%29%2F-.97

ria23 · Answer

Aha! Ok... I'm caught back up. :D
So yhu replaced the e with a natural log... 
And I would divide 1/9711 by -.97?

anonymous · Answer

take the log of both sides i.e. write in logarithmic form 
one you have $$\frac{1}{9711}=e^{-.97t}$$ you have to get $t$ out of the exponent 
you do that via 
$$\ln(\frac{1}{9711})=-.97t$$

anonymous · Answer

then simple algebra to get $t$ namely divide both sides by $-.97$

anonymous · Answer

i sent you the wolfram link with what i am pretty sure is the answer, typo free

anonymous · Answer

gotta run, good luck

ria23 · Answer

Ok n.n I kind of understand how this was worked out. c:
Thank yhu Satellite ^.^