Question
The power generated by an electrical circuit (in watts) as a function of its current
\[x\] (in amperes) is modeled by:
\[P(x)=-12x^2+120x\]
What current will produce the maximum power?
amperes
\[x\] (in amperes) is modeled by:
\[P(x)=-12x^2+120x\]
What current will produce the maximum power?
amperes
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To find the current \(x\) that produces maximum power in the function \(P(x) = -12x^2 + 120x\), we need to determine the vertex of the parabola represented by this quadratic equation. The standard form of a quadratic function is given by:
\[
P(x) = ax^2 + bx + c
\]
In this case:
- \(a = -12\)
- \(b = 120\)
The x-coordinate of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Substituting in the values of \(a\) and \(b\):
\[
x = -\frac{120}{2 \cdot -12} = -\frac{120}{-24} = 5
\]
Thus, the current that will produce the maximum power is
\[
\boxed{5} \text{ amperes}.
\]
\[
P(x) = ax^2 + bx + c
\]
In this case:
- \(a = -12\)
- \(b = 120\)
The x-coordinate of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Substituting in the values of \(a\) and \(b\):
\[
x = -\frac{120}{2 \cdot -12} = -\frac{120}{-24} = 5
\]
Thus, the current that will produce the maximum power is
\[
\boxed{5} \text{ amperes}.
\]
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