Answer:
[tex]P = \frac{1}{256}[/tex]
Step-by-step explanation:
Given
[tex]p=4^{-0.02t}[/tex]
Required
Determine P when t = 200
To do this, we simply substitute 200 for t in [tex]p=4^{-0.02t}[/tex]
[tex]p=4^{-0.02*200}[/tex]
[tex]P = 4^{-4}[/tex]
Apply law of indices: [tex]a^{-b} = \frac{1}{a^b}[/tex]
So:
[tex]P = 4^{-4}[/tex] becomes
[tex]P = \frac{1}{4^4}[/tex]
[tex]P = \frac{1}{256}[/tex]
As a decimal
[tex]P = 0.003906[/tex] (approximated)
Applying the formula, it is found that the fractional part of the batteries is still operating after 200 hours of use is of 0.003906.
The fractional part of batteries still working after t hours is an exponential function modeled by:
[tex]P(t) = 4^{-0.02t}[/tex]
Then, after 200 hours, we have have that [tex]t = 200[/tex], and hence:
[tex]P(200) = 4^{-0.02(200)}[/tex]
[tex]P(200) = 4^{-4}[/tex]
[tex]P(200) = \frac{1}{4^{4}}[/tex]
[tex]P(200) = 0.003906[/tex]
The fractional part of the batteries is still operating after 200 hours of use is of 0.003906.
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