b. After about 1,000 charging cycles, smartphone batteries lose about one fifth of

Question

their charging capacity. A brand new phone is currently at 78%. What would
this phone’s battery percentage show after 1,000 charging cycles? After 500?

Answer 1

To calculate the battery percentage after 500 and 1,000 charging cycles, we must first understand how the capacity degrades over these cycles.

Given that after about 1,000 charging cycles, smartphone batteries lose about one-fifth (or 20%) of their charging capacity, we can work through the calculations:

1. Initial Battery Percentage: 78%

2. Battery Decrease After 1,000 Charging Cycles:

   • The battery will lose 20% of its original capacity after 1,000 cycles.
   • Thus, the remaining capacity will be: \[ \text{Remaining Capacity} = \text{Initial Capacity} - (\text{Initial Capacity} \times 0.20) \] \[ = 78% - (78% \times 0.20) \] \[ = 78% - 15.6% \] \[ = 62.4% \]

3. Battery After 500 Charging Cycles:

   • If the battery loses its capacity gradually over the cycles, we can find the percentage lost after 500 cycles, which is half of 1,000 cycles.
   • Therefore, the energy lost after 500 cycles is approximately half of the loss after 1,000 cycles: \[ \text{Loss after 500 cycles} = 0.5 \times 15.6% = 7.8% \] \[ \text{Remaining Capacity after 500 cycles} = 78% - 7.8% = 70.2% \]

Summary:

• After 1,000 charging cycles, the battery percentage will be approximately 62.4%.
• After 500 charging cycles, the battery percentage will be approximately 70.2%.