To determine when the minimum power production occurs and what the minimum power production is, we need to analyze the given quadratic function [tex]\( P = h^2 - 12h + 210 \)[/tex], which represents the power produced in megawatts, where [tex]\( h \)[/tex] is the hour of the day.
### Step-by-Step Solution:
1. Identify the Form of the Quadratic Equation:
The given function [tex]\( P = h^2 - 12h + 210 \)[/tex] is a quadratic equation of the form [tex]\( P = ah^2 + bh + c \)[/tex] where:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -12 \)[/tex]
- [tex]\( c = 210 \)[/tex]
2. Determine the Vertex of the Parabola:
For a quadratic equation [tex]\( ah^2 + bh + c \)[/tex], the vertex represents the maximum or minimum point of the parabola. To locate the vertex, we use the formula for the hour [tex]\( h \)[/tex] at which this occurs:
[tex]\[
h_{\text{min}} = -\frac{b}{2a}
\][/tex]
Substituting in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
h_{\text{min}} = -\frac{-12}{2 \times 1} = \frac{12}{2} = 6
\][/tex]
3. Evaluate the Minimum Power Production:
Having determined that the minimum power production occurs at [tex]\( h = 6 \)[/tex], we need to calculate [tex]\( P \)[/tex] at this hour by substituting [tex]\( h = 6 \)[/tex] back into the original quadratic equation:
[tex]\[
P = (6)^2 - 12(6) + 210
\][/tex]
Calculate each term step-by-step:
[tex]\[
6^2 = 36
\][/tex]
[tex]\[
12 \times 6 = 72
\][/tex]
[tex]\[
P = 36 - 72 + 210
\][/tex]
[tex]\[
P = -36 + 210
\][/tex]
[tex]\[
P = 174
\][/tex]
Thus, the minimum power production occurs at [tex]\( h = 6 \)[/tex] hours, which is 6:00 AM, and the minimum power produced is [tex]\( 174 \)[/tex] megawatts.
