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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)=-12x^(2)+120 x What is the maximum power generated by the circuit? watts

The power generated by an electrical circuit (in watts) as a function of its current xx (in amperes) is modeled by:\newlineP(x)=−12x2+120xP(x)=-12x^{2}+120x\newlineWhat is the maximum power generated by the circuit?\newlinewatts\text{watts}

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Q. The power generated by an electrical circuit (in watts) as a function of its current xx (in amperes) is modeled by:\newlineP(x)=−12x2+120xP(x)=-12x^{2}+120x\newlineWhat is the maximum power generated by the circuit?\newlinewatts\text{watts}
  1. Identify Parabola Direction: To find the maximum power generated by the circuit, we need to find the vertex of the parabola represented by the quadratic function P(x)=−12x2+120xP(x) = -12x^2 + 120x. Since the coefficient of x2x^2 is negative, the parabola opens downwards, and the vertex will give us the maximum value of P(x)P(x).
  2. Calculate Vertex: The vertex of a parabola given by the function f(x)=ax2+bx+cf(x) = ax^2 + bx + c is at the point (h,k)(h, k), where h=−b2ah = -\frac{b}{2a}. In our case, a=−12a = -12 and b=120b = 120.
  3. Find x-coordinate of Vertex: Calculate the x-coordinate of the vertex hh using h=−b2ah = -\frac{b}{2a}.h=−1202×−12h = -\frac{120}{2 \times -12}h=−120−24h = -\frac{120}{-24}h=5h = 5
  4. Calculate y-coordinate of Vertex: Now we need to calculate the y-coordinate of the vertex kk, which is the maximum power P(h)P(h). We do this by substituting x=hx = h into the function P(x)P(x).P(h)=−12h2+120hP(h) = -12h^2 + 120h
  5. Substitute xx into Function: Substitute h=5h = 5 into P(h)P(h) to find the maximum power.\newlineP(5)=−12(5)2+120(5)P(5) = -12(5)^2 + 120(5)\newlineP(5)=−12(25)+600P(5) = -12(25) + 600\newlineP(5)=−300+600P(5) = -300 + 600\newlineP(5)=300P(5) = 300
  6. Calculate Maximum Power: The maximum power generated by the circuit is 300300 watts.

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