7. Case study question

For one particular jet, the height of the water above the water surface is given by

\(p(x)=-x^2+10x-24\)

where \(x\) is the horizontal distance in metres from a reference point and \(p(x)\) is in metres.

The water stream meets the water surface where \(p(x)=0\). The water surface itself lies \(0.1\text{ m}\) above the garden floor, resting on the supporting fountain rods.

Let \(\alpha\) and \(\beta\), with \(\alpha<\beta\), be the zeroes of \(p(x)\).

Based on the above information, answer the following:

1. Without solving for \(\alpha\) and \(\beta\) separately, find the horizontal width of the arc of water.

• 0
• −1
• 12
• 2

2. Find the horizontal position at which the stream reaches its greatest height, and find that greatest height above the water surface.

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• 7
• 11
• 6

3. Find \(H\), the maximum height of the stream measured from the garden floor.

• 1
• 4.1
• 1.1
• 1.2

4. A second jet is to meet the water surface at the same two points, but must reach a maximum height of \(3\text{ m}\) above the water surface. Determine