Aliens have injected Ana with mathematical nanobots. These nanobots force Ana to think about math more and more. Initially, there were only five nanobots. The population of the nanobots doubles every hour. The population of the nanobots follows the equations p(t) = 5·2^t. After there are 106nanobots, Ana will cease thinking of anything other than math. How many hours will it take for math to take over Ana's brain?

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elcharly64 elcharly64 · Answer 1 · 2020-02-14T05:00:48+01:00

Answer:

Math will take over Ana's brain at 4.4 hours

Step-by-step explanation:

Exponential Grow

The population of the nanobots follows the equation

[tex]p(t) = 5\cdot 2^t[/tex]

We must find the value of t such that the population of nanobots is 106 or more, that is

[tex]5\cdot 2^t\geq 106[/tex]

We'll solve the equation

[tex]5\cdot2^t= 106[/tex]

Dividing by 5

[tex]2^t= 106/5=21.2[/tex]

Taking logarithms

[tex]log(2^t)= log(21.2)[/tex]

By logarithms property

[tex]t\cdot log(2)= log(21.2)[/tex]

Solving for t

[tex]\displaystyle t=\frac{log21.2} {log2}[/tex]

[tex]t=4.4 \ hours[/tex]

Math will take over Ana's brain at 4.4 hours