Need help finding the Max Power in a math/EE problem please

[DominoK]

Homework Statement

The voltage across the inductor is given by v(t)=L(di(t)/dt).
Determine:

(1.) Tthe voltage v(t)
(2.) The power p(t)=v(t)*i(t).
(3.) The maximum power transferred ( t = ? (in seconds) when power is at maximum).

if the inductance is L=2 and the current i(t) is given by:

(a.) i(t)=13e^-200t A.
(b.) i(t)=20cos(2π60t) A.

It's the formula/steps to get p_max that I am stuck on.

Thank you so much!

Homework Equations

See above.

The Attempt at a Solution

I got as far as answering the (1.) and (2.) for each (a.) and (b.), but I do not know how to calculate the p_max.

(a1.) v(t) = -5.2 e^-200t V.
(a2.) p(t) = -67.6 e^-400t W.
(a3.) ? ( t = ? when power is at maximum).

(b1.) v(t) = -4.8 π sin(120πt) V.
(b2.) p(t) = -48sin(240πt) W.
(b3.) ? ( t = ? when power is at maximum).

Thank you very much!

 

[Mark44]

Homework Statement

The voltage across the inductor is given by v(t)=L(di(t)/dt).
Determine:

(1.) Tthe voltage v(t)
(2.) The power p(t)=v(t)*i(t).
(3.) The maximum power transferred ( t = ? (in seconds) when power is at maximum).

if the inductance is L=2 and the current i(t) is given by:

(a.) i(t)=13e^-200t A.
(b.) i(t)=20cos(2π60t) A.

It's the formula/steps to get p_max that I am stuck on.

Thank you so much!

Homework Equations

See above.

The Attempt at a Solution

I got as far as answering the (1.) and (2.) for each (a.) and (b.), but I do not know how to calculate the p_max.

(a1.) v(t) = -5.2 e^-200t V.
(a2.) p(t) = -67.6 e^-400t W.
(a3.) ? ( t = ? when power is at maximum).

What's the behavior of y = e^-400t? This is a decaying exponential function, which tells you something about where its maximum value is.

The negative sign on -67.6 looks wrong to me. Power should be nonnegative, I believe.

(b1.) v(t) = -4.8 π sin(120πt) V.
(b2.) p(t) = -48sin(240πt) W.
(b3.) ? ( t = ? when power is at maximum).

What does the graph of y = sin(240πt) look like? What is its maximum value, and where does it attain this maximum? Also, I'm not sure about the sign you have on -48.