The path of water from a hose on a fire tugboat can be approximated by the equation
y = −0.006x2 + 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.
just solve the quadratic using the quadratic formula:
−0.006x2 + 1.2x + 10 = 6
−0.006x2 + 1.2x + 4 = 0
and finish it off...
You will get two answers, but only one is appropriate.
To find the distance from the tugboat when the water from the hose is 6 feet above the ocean, we need to solve the equation y = -0.006x^2 + 1.2x + 10 for x when y = 6.
Substitute y = 6 into the equation: 6 = -0.006x^2 + 1.2x + 10.
Rearrange the equation to form a quadratic equation: -0.006x^2 + 1.2x + 10 - 6 = 0.
Combine like terms: -0.006x^2 + 1.2x + 4 = 0.
To solve the quadratic equation, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
In this case, a = -0.006, b = 1.2, and c = 4.
Substituting the values into the formula: x = (-1.2 ± √(1.2^2 - 4(-0.006)(4))) / (2(-0.006)).
Simplify the equation: x = (-1.2 ± √(1.44 + 0.096)) / (-0.012).
x = (-1.2 ± √1.536) / -0.012.
Taking the square root: x = (-1.2 ± 1.239) / -0.012.
Simplify further: x = -102.5 or x = 62.5.
The distance from the tugboat when the water from the hose is 6 feet above the ocean is approximately 62.5 feet, rounded to the 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 equation y = 6 for x.
The given equation is y = -0.006x^2 + 1.2x + 10.
Substituting y = 6 into the equation, we get:
6 = -0.006x^2 + 1.2x + 10
Rearranging the equation, we get:
0 = -0.006x^2 + 1.2x + 4
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = -0.006, b = 1.2, and c = 4.
Plugging these values into the quadratic formula, we get:
x = (-1.2 ± √(1.2^2 - 4*(-0.006)*4)) / (2*(-0.006))
Simplifying further:
x = (-1.2 ± √(1.44 + 0.096)) / (-0.012)
x = (-1.2 ± √1.536) / (-0.012)
Now we can calculate the two possible solutions for x:
x1 = (-1.2 + √1.536) / (-0.012)
x2 = (-1.2 - √1.536) / (-0.012)
Using a calculator, x1 ≈ -14.36 and x2 ≈ 174.36.
Since we are interested in the distance, which cannot be negative, the distance from the tugboat when the water from the hose is 6 feet above the ocean is approximately 174.36 feet.