The resistance of a certain component on the Enterprise decreases as the current

Question

The resistance of a certain component on the Enterprise decreases as the current through it increases, as described by the relation
R= 80/( 12 + 4i^3),

a. Determine the current that results in maximum power in the unit

b. and the maximum power delivered to the unit.
HINT you have R(i) : maximum power will be for the i that maximizes R.
How do you get the maximum of function R(i) and the current that will do it? That current will give you maximum power Im we recall P=(im^2R (im) Reminder R will change with i in this case!

Explanation / Answer

a)   Here,    R= 80/( 12 + 4i3)

=>   maximum power will be for the i that maximizes ,P = 80i2/( 12 + 4i3)

Differentiating and equating to zero .

=>    Maximum value of R will be when    i3    = 6

=>    i   =   1.817 A

=>    current that results in maximum power in the unit   =   1.817 A

b)     the maximum power delivered to the unit   = i2 *R

                                                                            = (1.817 )2 * (80/36)

                                                                         = 7.336 W

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