Question

The power generated by an electrical circuit (in watts) as a function of its current x xx (in amperes) is modeled by P ( x ) = − 15 x ( x − 8 ) P(x)=−15x(x−8)P, left parenthesis, x, right parenthesis, equals, minus, 15, x, left parenthesis, x, minus, 8, right parenthesis What current will produce the maximum power?

Answer 1

Given:

The power generated by an electrical circuit (in watts) as a function of its current x (in amperes) is modeled by

$P(x)=-15x(x-8)$

To find:

The current which will produce the maximum power.

Solution:

We have,

$P(x)=-15x(x-8)$

$P(x)=-15x^2+120x$

Differentiate with respect to x.

$P'(x)=-15(2x)+120(1)$

$P'(x)=-30x+120$      ...(i)

To find the extreme point equate P'(x)=0.

$-30x+120=0$

$-30x=-120$

Divide both sides by -30.

$x=4$

Differentiate (i) with respect to x.

$P'(x)=-30(1)+(0)$

$P'(x)=-30$ (Maximum)

It means, the given function is maximum at x=4.

Therefore, the current of 4 amperes will produce the maximum power.