Question

Batteries are rated in terms of ampere-hours (A·h). For example, a battery that can produce a current of 2.00 A for 3.00 h is rated at 6.00 A·h.

(a) What is the total energy stored in a 9.0 V battery rated at 47.0 A·h?

(b) At $0.0660 per kilowatt-hour, what is the value of the electricity produced by this battery? NOTE: Enter your response with hundredths precision.

Answer 1

(a) 423 J

The power of the battery is the ratio between the total energy stored (E) and the time elapsed (t):

$P=\frac{E}{t}$

However, the power is also the product of the voltage (V) and the current (I):

$P=VI$

Linking the two equations together,

$\frac{E}{t}=VI\\E=VIt$

Since we know:

V = 9.0 V

$I \cdot t = 47.0 A\cdot h$

We can calculate the total energy:

$E=(9.0 V)(47 A \cdot h)=423 J$

(b) $7.79\cdot 10^{-6}$ dollars

The battery has a total energy of E = 423 J. (2)

1 Watt (W) is equal to 1 Joule (J) per second (s):

$1 W = \frac{1 J}{1 s}$

so 1 kW corresponds to 1000 J/s:

$1 kW = \frac{1000 J}{1 s}$

Multiplying both side by 1 hour (1 h):

$1 kW \cdot h = \frac{1000 J}{1 s} 1 h$

and $1 h = 3600 s$, so

$1 kWh = \frac{1000 J}{1 s}\cdot 3600 s =3.6\cdot 10^6 J$

So we find the conversion between kWh and Joules. So now we can convert the energy from Joules (2) into kWh:

$1 kWh = 3.6\cdot 10^6 J = x : 423 J\\x=\frac{1 kWh \cdot 423 J}{3.6\cdot 10^6 J}=1.18\cdot 10^{-4}kWh$

And since the cost is $0.0660 per kilowatt-hour, the total cost will be

$C= 0.0660\cdot 1.18\cdot 10^{-4} kWh=7.79\cdot 10^{-6}$ dollars