Getting Power from Voltage and Current

[Temp0]

Homework Statement

A constant current of 3 A for 4 hours is required to charge an automotive
battery, and the battery's terminal voltage is v(t) = 10 + t/2 V, where t is in hours.
Assuming an electricity cost $0.12 per KWh, what is the cost to charge the battery?

Homework Equations

p(t) = i(t)v(t)

The Attempt at a Solution

So I just thought that I could multiply current (3A) with the voltage (10 + 4h/2)V and get power in watts, which would be 36W. I divide 36 by 1000, 0.036kW, and then I multiply that by the cost ($0.12/kWh) to get some number/h, and then multiply that number by 4 hours to get the cost to charge the battery. However, this isn't giving me the right answer, I would greatly appreciate it if someone could point me in the right direction. Thank you.

 

[SteamKing]

The electricity cost is metered by the total amount of energy charged into the battery (Notice: the price of electricity is $0.12 per kilowatt-hour).

You have a changing voltage as the battery charges but a constant current input. What would be the total energy required to charge the battery according to the info in the OP?

 

[Temp0]

Well, I know that energy is the integral of p(t), so would I take the integral with the lower limit being 0 and upper limit being 4, and then multiply that number by the cost of electricity?

 

[SteamKing]

yes.

 

[Temp0]

Alright, thanks a lot for your help. Just to clarify, I took $0.012/kWh, multiplied it by 1kWh/3600000J, and then I got $number/J, and then I calculated for the amount of joules using the integral of my power equation, substituting in 4 for time, and multiplied that number by the price of electricity/joules.