# helpin a 120-volt circuit having a resistance of 12 ohms, the power W in watts when a current I is flowing through is given by W=120 I - 12 I^2. what is the maximum power, in watts, that can be...

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in a 120-volt circuit having a resistance of 12 ohms, the power W in watts when a current I is flowing through is given by W=120 I - 12 I^2. what is the maximum power, in watts, that can be delivered in this circuit?

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w = 120 I - 12 i^2

To find the maximum value, we will need to find the derivativ'es zero.

==> w' = 120 - 24 I = 0

==> I = 120/24 = 5

**Since the factor of I^2 is negative, then the function has Maximum value when I = 5**

==> w = 120*I - 12 I^2

= 120*5 - 12*5^2

= 600 - 300 = 300

**Then the maximum power is W = 300 watts.**

The equation power w is given by :

W = 120i - 12i^2

To find the maximum power W.

To maximise W:

W/12 = (120i-12i^2)/12 = 10i-i^2

W/12 = 2*5i-i^2

W/12 =5^5 -5^2+2*5i-i^2

W/12 = 5^2 - (5 - i)^2

W/12 <5^2 , as (5-i)^2 being a perfect square is always positive.

So the W/12 is maximum and equal to 5^2=25 when i = 5 amp.

Or the maximum power that could drawn is W = 25*12 = 300 watts, when i = 5amp.