Answer:
(a) The exponential function representing the number of people who had watched the video
hours after the initial observation is
.
(b) As
, we conclude that this video is not going "viral".
Step-by-step explanation:
Statement is incomplete. The complete statement is:
<em>An internet analytics company measured the number of people watching a video posted on a social media platform. The company found 129 people had watched the video and that the number of people who had watched it was increasing by 30% every 3 hours. </em>
<em>(a)</em><em> Write an exponential function for the number of people A who had watched the video n hours after the initial observation.</em>
<em>(b)</em><em> A video is said to go "viral" if the number of people who have watched the video exceeds 5 million within 5 days (120 hours). Would this video be considered to have gone viral?</em>
(a) From the statement of the problem we get the following relationship:
(Eq. 1)
Where:
- i-th number of people watching a video, dimensionless.
- (i+1)-th number of people watching a video, dimensionless.
- Increase ratio, dimensionless.
For Induction Theorem, we get the following relatioship for
:
(Eq. 2)
For
, the following relationship is constructed:
(Eq. 3)
And for
, we have the following expression:
(Eq. 4)
If we multiply (Eq. 3) by the (m+1)-th ratio based on (Eq. 1):
![\left(\frac{n_{m+1}}{n_{o}}\right)\cdot \left(\frac{n_{m+2}}{n_{m+1}} \right) = r^{m} \cdot r](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7Bn_%7Bm%2B1%7D%7D%7Bn_%7Bo%7D%7D%5Cright%29%5Ccdot%20%5Cleft%28%5Cfrac%7Bn_%7Bm%2B2%7D%7D%7Bn_%7Bm%2B1%7D%7D%20%5Cright%29%20%3D%20%20r%5E%7Bm%7D%20%5Ccdot%20r)
![\frac{n_{m+2}}{n_{o}} = r^{m+1}](https://tex.z-dn.net/?f=%5Cfrac%7Bn_%7Bm%2B2%7D%7D%7Bn_%7Bo%7D%7D%20%3D%20r%5E%7Bm%2B1%7D)
Which is (Eq. 4) and the exponential function is represented by:
,
(Eq. 5)
As the number of people is increased at constant rate every 3 hours, we get that
is:
,
,
(Eq. 6)
Where
is the time, measured in hours.
Then, the exponential function is:
(Eq. 7)
Where
is the initial number of people watching the video, dimensionless.
If we know that
and
, then the exponential function representing the number of people who had watched the video
hours after the initial observation is:
![n(t) = 129\cdot 1.3^{\frac{t}{3} }](https://tex.z-dn.net/?f=n%28t%29%20%3D%20129%5Ccdot%201.3%5E%7B%5Cfrac%7Bt%7D%7B3%7D%20%7D)
(b) If we know that
, then we evaluate the exponential function:
![n (120) = 129\cdot 1.3^{\frac{120}{3} }](https://tex.z-dn.net/?f=n%20%28120%29%20%3D%20129%5Ccdot%201.3%5E%7B%5Cfrac%7B120%7D%7B3%7D%20%7D)
![n(120) = 4.659\times 10^{6}](https://tex.z-dn.net/?f=n%28120%29%20%3D%204.659%5Ctimes%2010%5E%7B6%7D)
As
, we conclude that this video is not going "viral".