how high above the river is the water arc?

Question

on one of the banks f the chicago river, there is a water cannon, called the water arc, that sprays recirculated water across the rivr. the path of the water arc is given by the model y=0.006x^2+1.2+10 where x is the distance (in feet) across the river and y is the height f the arc (in feet)

Answer 1

To find how high above the river the water arc is, we need to determine the maximum height of the water arc given by the model 𝑦=−0.006𝑥2+1.2𝑥+10y=−0.006x2+1.2x+10.

Step-by-Step Solution:

1. Identify the Model: The equation given is:

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y = -0.006x^2 + 1.2x + 10

1. This is a quadratic equation of the form 𝑦=𝑎𝑥2+𝑏𝑥+𝑐y=ax2+bx+c, where:
2. 𝑎=−0.006a=−0.006
3. 𝑏=1.2b=1.2
4. 𝑐=10c=10
5. Find the Vertex: The vertex of a parabola given by 𝑦=𝑎𝑥2+𝑏𝑥+𝑐y=ax2+bx+c is at 𝑥=−𝑏/(2𝑎)x=−b/(2a).
6. Calculate the x-coordinate of the Vertex:

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x = -1.2 / (2 * -0.006)

x = -1.2 / -0.012

x = 100

1. Find the y-coordinate of the Vertex: Substitute 𝑥=100x=100 back into the original equation to find the height of the arc.

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y = -0.006(100)^2 + 1.2(100) + 10

y = -0.006(10000) + 120 + 10

y = -60 + 120 + 10

y = 70

Result

The maximum height of the water arc is 70 feet above the river.