Big ideas chapter 8 algebra 2

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

First, we need to recognize this is an example of a sum of infinite series. Based on the question, it is assumed that this converges.

What the question is asking is that the battery is losing 0.02 percent charge each time it charges. This can be represented by the following equation where sequential entities are added to 20 after being sequentially multiplied by 0.98.

1 2 3 4

20 + [0.98(20)] + [0.98*(0.98(20))] + [0.98 * (0.98*(0.98(20)))]...

At some point, the battery life will be 0 and we are tasked with finding the total hours of the battery. This can be written as the following formula:

_∞

∑a(r^k)

k = 0

which is read as the sum of a times the common ration to the k power where a is the first value, k is the term, and r is the ratio. Now there is good news. Turns the sum of the infinite geometric series has an equation as well:

_∞

∑a(r^k) = a/(1-r)

k = 0

So, now we will plug in everything we know.

20/ (1 - 0.98) = 20 / 0.02 = 2000/2 = 1000

The battery will have 1000 hours to it if it holds only 98% of the charge each time. .

 

Answer 2

This is an infinite geometric series

Life = 20 / (1 - 0.98)

Life = 20 / 0.02)

Life = 1000