We are given the quadratic function [tex]\( D(h) = -0.078h^2 + 3.811h - 32.433 \)[/tex]. Our task is to analyze the behavior of this function by finding its critical points and roots.
### 1. Vertex of the Quadratic Function
The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found using the vertex formula for [tex]\( x \)[/tex]:
[tex]\[ h_{\text{vertex}} = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = -0.078 \)[/tex] and [tex]\( b = 3.811 \)[/tex]. Substituting these values into the formula gives us:
[tex]\[ h_{\text{vertex}} = -\frac{3.811}{2 \times -0.078} = \frac{3.811}{0.156} \approx 24.429 \][/tex]
Next, we need to find the value of [tex]\( D(h) \)[/tex] at this vertex:
[tex]\[ D(h_{\text{vertex}}) = -0.078(24.429)^2 + 3.811(24.429) - 32.433 \][/tex]
Evaluating this expression, we get:
[tex]\[ D(h_{\text{vertex}}) \approx 14.117 \][/tex]
Thus, the vertex of the function is at [tex]\( (24.429, 14.117) \)[/tex]. This indicates that the maximum value of [tex]\( D(h) \)[/tex] is 14.117 when [tex]\( h = 24.429 \)[/tex].
### 2. Roots of the Quadratic Function
The roots of the quadratic function can be found by solving the equation [tex]\( D(h) = 0 \)[/tex]:
[tex]\[ -0.078h^2 + 3.811h - 32.433 = 0 \][/tex]
To find the roots, we can use the quadratic formula:
[tex]\[ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For our function, [tex]\( a = -0.078 \)[/tex], [tex]\( b = 3.811 \)[/tex], and [tex]\( c = -32.433 \)[/tex]. Substituting these values into the quadratic formula, we get:
[tex]\[ h = \frac{-3.811 \pm \sqrt{3.811^2 - 4(-0.078)(-32.433)}}{2(-0.078)} \][/tex]
Solving this equation gives us two roots:
[tex]\[ h \approx 10.976 \][/tex]
[tex]\[ h \approx 37.883 \][/tex]
### Summary
- The vertex of the quadratic function [tex]\( D(h) = -0.078h^2 + 3.811h - 32.433 \)[/tex] is at [tex]\( h = 24.429 \)[/tex] and [tex]\( D(h) = 14.117 \)[/tex].
- The roots of the function are approximately [tex]\( h = 10.976 \)[/tex] and [tex]\( h = 37.883 \)[/tex].
This completes the detailed analysis of the given quadratic function.
