To find the height of the highest point of the archway modeled by the function [tex]\( h(x) = -x^2 - 10x + 5 \)[/tex], we will determine the vertex of this quadratic function.
A quadratic function in the form [tex]\( h(x) = ax^2 + bx + c \)[/tex] achieves its maximum or minimum value at the vertex. For a downward-opening parabola (where [tex]\( a < 0 \)[/tex]), the vertex represents the maximum point on the graph.
The coordinates of the vertex, [tex]\((x, h(x))\)[/tex], can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = \frac{-b}{2a} \][/tex]
In our function [tex]\( h(x) = -x^2 - 10x + 5 \)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = -10 \)[/tex]
- [tex]\( c = 5 \)[/tex]
Plug these values into the vertex formula:
[tex]\[ x = \frac{-(-10)}{2(-1)} = \frac{10}{-2} = -5 \][/tex]
This x-coordinate, [tex]\( x = -5 \)[/tex], represents the horizontal distance from one side of the arch support to the other.
Next, we substitute [tex]\( x = -5 \)[/tex] back into the original equation to find the corresponding height [tex]\( h(x) \)[/tex]:
[tex]\[ h(-5) = -(-5)^2 - 10(-5) + 5 \][/tex]
Calculate each term:
[tex]\[ -(-5)^2 = -25 \][/tex]
[tex]\[ -10(-5) = 50 \][/tex]
[tex]\[ + 5 = 5 \][/tex]
Combine these results to find [tex]\( h(-5) \)[/tex]:
[tex]\[ h(-5) = -25 + 50 + 5 = 30 \][/tex]
Thus, the height of the highest point of the archway is:
[tex]\[ 30 \text{ feet} \][/tex]
The vertex occurs at [tex]\( x = -5 \)[/tex] and the maximum height [tex]\( h(x) \)[/tex] at this point is 30 feet. Hence, the height of the highest point of the archway is 30 feet.
