The path of water from a hose on a fire tugboat can be approximated by the equation
y = −0.0055x2 + 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.
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To find the distance from the tugboat when the water is 6 feet above the ocean, we need to solve the equation for x when y = 6.
Given equation: y = -0.0055x^2 + 1.2x + 10
Substituting y = 6 into the equation:
6 = -0.0055x^2 + 1.2x + 10
Rearranging the equation:
0 = -0.0055x^2 + 1.2x + 10 - 6
0 = -0.0055x^2 + 1.2x + 4
Now we have a quadratic equation in standard form (ax^2 + bx + c = 0), where a = -0.0055, b = 1.2, and c = 4.
To solve the quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values:
x = (-(1.2) ± √((1.2)^2 - 4(-0.0055)(4))) / (2(-0.0055))
Simplifying:
x = (-1.2 ± √(1.44 + 0.088)) / (-0.011)
x = (-1.2 ± √1.528) / (-0.011)
Calculating the square root:
x = (-1.2 ± 1.236) / (-0.011)
Using the positive square root:
x = (-1.2 + 1.236) / (-0.011)
x = 0.036 / (-0.011)
x ≈ -3.27
Rounded to the nearest hundredth, the distance from the tugboat when the water is 6 feet above the ocean is approximately -3.27 feet.
Please note that negative distances in this context may not have any physical meaning, so it's important to consider the validity and interpretation of such values.