The path of water from a hose on a fire tugboat can be approximated by the equation

y = −0.0055x^2 + 1.2x + 10,
where y is the height, in feet, of the water above the ocean when the water is x feet from the tugboat. When the water from the hose is 6 feet above the ocean, at what distance from the tugboat is it? Round answer to nearest hundredth.

To find the distance from the tugboat when the water from the hose is 6 feet above the ocean, we need to solve the given equation for x.

The equation provided is y = -0.0055x^2 + 1.2x + 10, where y represents the height of the water above the ocean and x represents the distance from the tugboat.

Given that the height of the water (y) is 6 feet, we can substitute this value into the equation:

6 = -0.0055x^2 + 1.2x + 10

Rearranging the equation, we get:

0.0055x^2 - 1.2x - 4 = 0

Now, we can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula. In this case, it is easier to use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 0.0055, b = -1.2, and c = -4.

Plugging in these values, we get:

x = (1.2 ± √((-1.2)^2 - 4(0.0055)(-4))) / (2(0.0055))

Simplifying further:

x = (1.2 ± √(1.44 + 0.088)) / 0.011

x = (1.2 ± √1.528) / 0.011

Taking the square root and dividing:

x ≈ (1.2 ± 1.236) / 0.011

Now, we can calculate the two possible values for x:

1. x ≈ (1.2 + 1.236) / 0.011 ≈ 201.09
2. x ≈ (1.2 - 1.236) / 0.011 ≈ -3.09

Since we are dealing with a distance, we can ignore the negative value, so the distance from the tugboat when the water is 6 feet above the ocean is approximately 201.09 feet. Rounded to the nearest hundredth, the answer is approximately 201.09 feet.

nothing tough about this one. Just solve

−0.0055x^2 + 1.2x + 10 = 6
you should get two answers, but only one makes sense here, since the water starts out at 10 ft above the ocean.