According to one source, the amount of data passing through mobile pho

FobelloE 2021-08-11 Answered
According to one source, the amount of data passing through mobile phone networks doubles each year.
Explain why the amount of data passing through mobile phone networks is an exponential function of time. The amount of data grows by a factor of_________________ each year, this is an exponential function because the amount is growing by _____________________ multiples.
Use Do for the initial amount of data and find a formula that gives the data D as an exponential function of the time t in years.
If this trend continues, how long will it be before the amount of data is 100 times its initial value? Round to three decimal places.
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Expert Answer

Sadie Eaton
Answered 2021-08-12 Author has 104 answers
Step 1
Given: The amount of data passing through mobile phone networks doubles each year.
Step 2
a) We have given that the amount of data passing through mobile phone networks doubles each year.
That is if we assume the initial amount of data is A0 and A(t) is amount of data after t years, then after one year the amount of data becomes A(1)=2A0.
After 2 years the amount of data becomes A(2)=2A(1)=2(2A0)=4A0.
After 3 years the amount of data becomes A(3)=2A(2)=2(4A0)=8A0.
Continuing in this way after n years we get, A(n)=2A(n1).
Therefore, as time increases by 1 year the amount of data is obtained by multiplying the previous amount by 2.
That is the amount of data growing by constant multiples with time. Therefore, the amount of data passing through mobile phone networks is an exponential function of time.
The exponential function is of the form A(t)=A0ekt where A(t) is amount of data after t years, A0 is the initial amount of data and k is a growth factor.
Now, we have to find the value of k.
After 1 year the amount of data becomes A(1)=2A0.
By using the exponential function we get,
2A0=A0ek(1)
2=ek
k=ln(2)0.6931
Therefore, the amount of data grows by a factor of ln(2)0.6931 each year, this is an exponential function because the amount is growing by 2(twice) multiples.
Step 3
b) To find: The exponential function using D0 as the initial amount of data and D as the amount after t years.
Use D0 as the initial amount of data and find a formula that gives the data D as an exponential function of the time t in years.
We get,
D(t)=D0eln(2)t=D0(eln(2))t=D0(2)t
Therefore, the exponential function is D(t)=2tD0.
Step 4
c) To find: How long will it be before the amount of data is 100 times its initial value if the trend continues.
Use D(t)=100D0 in the exponential function from part (b) we get,
100D0=2tD0
Solve for t,
100=2t
Apply log on both sides,
log(100)=log(2t)
2=tlog(2)
t=2log(2)6.644
Therefore, it will take 6.644 years to make the amount of data 100 times its initial value.
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