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The power generated by an electrical circuit (in watts) as a function of its current x (in amperes) is modeled by P(x)=-15 x(x-8) What current will produce the maximum power? ◻ amperes

The power generated by an electrical circuit (in watts) as a function of its current \newlinexx (in amperes) is modeled by\newlineP(x)=−15x(x−8)P(x)=-15 x(x-8)\newlineWhat current will produce the maximum power?\newline□\square amperes

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Q. The power generated by an electrical circuit (in watts) as a function of its current \newlinexx (in amperes) is modeled by\newlineP(x)=−15x(x−8)P(x)=-15 x(x-8)\newlineWhat current will produce the maximum power?\newline□\square amperes
  1. Analyze Quadratic Function: To find the current that will produce the maximum power, we need to analyze the quadratic function P(x)=−15x(x−8)P(x) = -15x(x - 8). This is a parabola that opens downwards because the coefficient of the x2x^2 term is negative (−15-15). The maximum power will be at the vertex of this parabola.
  2. Compare to General Form: The general form of a quadratic function is ax2+bx+cax^2 + bx + c. In our case, the function P(x)=−15x2+120xP(x) = -15x^2 + 120x can be compared to this form, where a=−15a = -15 and b=120b = 120. The xx-coordinate of the vertex of a parabola given by ax2+bx+cax^2 + bx + c is found using the formula −b/(2a)-b/(2a).
  3. Find x-coordinate of Vertex: We will apply the formula to find the x-coordinate of the vertex. For our function P(x)=−15x2+120xP(x) = -15x^2 + 120x, we have a=−15a = -15 and b=120b = 120. Plugging these values into the formula gives us x=−120(2⋅−15)x = \frac{-120}{(2 \cdot -15)}.
  4. Calculate x-coordinate: Calculating the x-coordinate of the vertex, we get x=−120(2⋅−15)=−120−30=4x = \frac{-120}{(2 \cdot -15)} = \frac{-120}{-30} = 4.
  5. Current for Maximum Power: The xx-coordinate of the vertex, which is 44 amperes, represents the current that will produce the maximum power in the electrical circuit.

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