Question

An irrigation system (sprinkler) has a parabolic pattern. Theheight, in feet, of the spray of water is given by the equationh(x) = -z + 10x + 9.5, where T is the number of feetaway from the sprinkler head (along the ground) the sprayis.The irrigation system is positionedfeet abovethe ground to start.The spray reaches aheight offeet at horizontal distance offeet away from thesprinkler head.The spray reaches all the way to the ground at aboutfeet away.

Answer 1

Given

height of the spray of water is given by

`h\mleft(x\mright)=-x^2+10x+9.5`

where , x is the number of feet away from the sprinkler head the spray is

Find

The irrigation system is positioned ----- feet above the ground to start.

The spray reaches a height of------- feet at horizontal distance of

feet away from the sprinkler head.

The spray reaches all the way to the ground at about ------ feet away.

Step-by-step explanation

to find the height above the ground to start,

we have to put x = 0

so , h(x) = 9.5

hence , the irrigation system is positioned 9.5 feet above the ground to start.

the axis of symmetry of quadratic is x = -b/2a

so ,for given quadratic ,

`x=-(10)/((-2))=5`

so , the maximum height is

`\begin{gathered} h(5)=-(5)^2+10(5)+9.5 \\ h(5)=-25+50+9.5 \\ h(5)=34.5 \end{gathered}`

hence , the spray reaches a height of 34.5 feet at horizontal distance of feet away from the sprinkler head.

the maximum distance will be

`\begin{gathered} √(34.5)+5 \\ 5.87367006224+5 \\ 10.8736700622\approx10.9 \end{gathered}`

hence , the spray reaches all the way to the ground at about 10.9

feet away.

Final Answer

Hence , The irrigation system is positioned 9.5 feet above the ground to start.

The spray reaches a height of 34.5 feet at horizontal distance of

feet away from the sprinkler head.

The spray reaches all the way to the ground at about 10.9 feet away.