Question

After each charging, a battery is able to hold only 98% of the charge from the previous charging. The battery was used for 20 hours on its first charge before it had to be recharged. What is the total number of hours the battery can be used over its lifetime?

Answer 1

Answer:

$t_{total}=985 h$

Explanation:

The time of use of the battery can de determined by:

$t=0.98^n*20h$

t= is the time of duration

n=is the number of recharges

Now, we need to calculate n for a t=0h

$t=0.98^n*20h$

$\frac{t}{20h}=0.98^n$

$ln(\frac{t}{20h})=ln(0.98^n)$

$ln(\frac{t}{20h})=n*ln(0.98)$

$n=ln\frac{ln(t/20h)}{ln(0.98)}$

We can apply the limit of t trending to 0 or we can stablish a value of t which we consider that the battery will be useless. For example t=0.1 h

$n=ln\frac{ln(0.1h/20h)}{ln(0.98)}$

$n=262$

For the total time:

$t_{total}= \int_{0}^{262} 20h *0.98^{n}*dn$

$t_{total}= 20h*(\frac{*0.98^{262}}{ln(0.98)}-\frac{*0.98^{0}}{ln(0.98)})$

$t_{total}=985 h$