Question

Please help me solve this short problem I’m really struggling

Answer 1

Explanation:

In the standard form of a quadratic,

`h(x)=-x^2+v_0x+h_0`,

that last term, the h0, is the initial height off the ground that the object is. That means that for the first question, the irrigation system is initially 9.5 feet off the ground.

In order to answer the next question, we need to find the vertex of the parabola, because this is where the max height will occur and at how many feet away from the system. The max height is the k of the vertex and the horizontal distance away is the h of the vertex: (h, k). Thankfully, we have simple formulas to find those:

`h=-(b)/(2a)` and
`k=c-(b^2)/(4a)` where a, b, and c come from the coefficients in the equation. For us, a = -1, b = 10, and c = 9.5. Finding h first:

`h=-(10)/(2(-1))=(-10)/(-2)=5`

and now k:

`k=9.5-((10^2)/(4(-1)))=9.5-((100)/(-4) )=9.5-(-25)=9.5+25=34.5`

Summing it up, the vertex is at (5, 34.5), The interpretation of that is that "The spray reaches a max height of 34.5 feet at a horizontal distance of 5 feet away from the sprinkler head."

Next we are asked how far away from the sprinkler head the water hits the ground. To do this we factor the expression to find the zeros. The zeros tell us where the water will hit the ground after it reaches its max height and then falls back down to earth.

I used my calculator to factor this for me to get that the water will hit the ground 10.87 feet away from the sprinkler head.