Question

The rate of change of charge is passing into a battery is modeled by the function C(t) = 10 + 6sin (t^2 / 3)#, the battery can hold a charge of 160 coulombs. How long, in hours, does it take to charge the battery?

Answer 1

See below.

Explanation:

Before addressing the question, we can agree, because it is solid physics, that if `Q` is the amount of charge in the battery, then the current flowing (around the circuit and "into" the battery) is `I = (dQ)/(dt)`.

I say that because I think your question might be unfortunately worded. At least it confuses me :)

`(dQ)/(dt)` is the rate of change of charge (with respect to time) in the battery.

So, on that basis, we can say this:

`color(red)((dQ)/(dt)) = I = C(t) = color(red)(10 + 6sin (t^2 / 3))`

And if that is the correct interpretation, then "all" we need to do is to solve this for `tau`, which is the time for the charge `Q` to become 160C:

`int_(t=0)^tau 10 + 6sin (t^2 / 3) dt = 160`

The next problem is that this cannot be solved using elementary functions.