Final answer:
The minimum power production of the power plant, given by the equation P = h² - 12h + 210, occurs at 6 AM with a production of 174 megawatts.
Step-by-step explanation:
Finding the Minimum Power Production
The given quadratic function representing the power generated by the power plant in megawatts is P = h² - 12h + 210, where h is the time in hours after midnight. To find the minimum power production and the corresponding time, we need to find the vertex of this parabola. This can be done using the vertex formula for a quadratic equation, which is h = -b/(2a), where a is the coefficient of h² and b is the coefficient of h.
For our equation, a = 1 and b = -12, so the time for minimum power production is h = -(-12)/(2×1), which simplifies to h = 6. Inserting this back into the equation, the minimum power production is P = 6² - 12×6 + 210, which calculates to P = 36 - 72 + 210 = 174 MW.
Therefore, the minimum power production occurs at 6 AM, and the minimum power produced is 174 megawatts.