Question

A car owner forgets to turn off the headlights of his car while it is parked in his garage. If the 12.0-V battery in his car is rated at 51.0 A · h and each headlight requires 27.0 W of power, how long will it take the battery to completely discharge?

Answer 1

Answer: 22.6 hours

Explanation:

The power is the measure of the rate of energy.

In this problem, the 12.0 V battery is rated at 51.0 Ah, which means it delivers 51.0 A of current in a time of t = 1 h = 3600 s. The power delivered by the battery can be written as

$P=IV$

where

I is the current

V = 12.0 V is the voltage of the battery

So the energy delivered by the battery can be written as

$E=Pt=VIt$

Where

$It=51.0 A\cdot h = 51.0 A \cdot 3600 s/h=183,600 A\cdot s$

So the energy delivered is

$E=(12.0)(183,600)=2.2\cdot 10^6 J$

At the same time, the headlight consumes 27.0 W of power, so 27 Joules of energy per second; Therefore, it will remain on for a time of:

$t=\frac{2.2\cdot 10^6 J}{27.0 W}=81481 s = 22.6 h$