To solve the problem, let's interpret and calculate key features step by step using the information from the given equation [tex]\( h(x) = -x^2 + 10x + 7.5 \)[/tex].
1. Starting Height:
- The irrigation system is positioned 7.5 feet above the ground to start.
This is evident from the constant term in the equation [tex]\( h(x) \)[/tex]. Therefore, the initial height is 7.5 feet.
2. Maximum Height:
- To find the maximum height, we need to find the vertex of the parabola. In a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex], the x-coordinate of the vertex can be found using the formula:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Here, [tex]\(a = -1\)[/tex] and [tex]\(b = 10\)[/tex]. Plugging these values into the formula gives:
[tex]\[
x = -\frac{10}{2(-1)} = 5
\][/tex]
So, the maximum height occurs at [tex]\( x = 5 \)[/tex] feet.
- To find the height at this point, substitute [tex]\( x = 5 \)[/tex] back into the equation:
[tex]\[
h(5) = -5^2 + 10(5) + 7.5 = -25 + 50 + 7.5 = 32.5
\][/tex]
Therefore, the maximum height of the spray is 32.5 feet.
3. Ground Reach:
- To determine the horizontal distance at which the spray reaches the ground, we need to solve for [tex]\( x \)[/tex] when [tex]\( h(x) = 0 \)[/tex]. This involves solving the quadratic equation:
[tex]\[
-x^2 + 10x + 7.5 = 0
\][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], with [tex]\( a = -1 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = 7.5 \)[/tex]:
[tex]\[
x = \frac{-10 \pm \sqrt{10^2 - 4(-1)(7.5)}}{2(-1)} = \frac{-10 \pm \sqrt{100 + 30}}{-2} = \frac{-10 \pm \sqrt{130}}{-2}
\][/tex]
Solving for both roots:
[tex]\[
x_1 = \frac{-10 + \sqrt{130}}{-2} \approx 10.7 \quad (\text{keeping the positive root})
\][/tex]
Therefore, the spray reaches the ground at approximately 10.7 feet away from the sprinkler head.
To summarize:
- The irrigation system is positioned 7.5 feet above the ground to start.
- The spray reaches a maximum height of 32.5 feet at a horizontal distance of 5 feet away from the sprinkler head.
- The spray reaches all the way to the ground at about 10.7 feet away.
