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 3 feet above the ocean, at what distance from the tugboat is it?
just solve for x in
−0.0055x2 + 1.2x + 10 = 3
To find the distance from the tugboat when the water is 3 feet above the ocean, we need to solve the equation y = 3 for x.
Given the equation: y = -0.0055x^2 + 1.2x + 10
Substituting y = 3, we get:
3 = -0.0055x^2 + 1.2x + 10
Rearranging the equation to standard quadratic form, we have:
0 = -0.0055x^2 + 1.2x + 7
Now we can solve this equation to find the value(s) of x.
Using the quadratic formula, x = (-b ± sqrt(b^2 - 4ac))/(2a), where a, b, and c are the coefficients in the quadratic equation.
In this case, a = -0.0055, b = 1.2, and c = 7.
Plugging these values into the quadratic formula, we get:
x = (-1.2 ± sqrt(1.2^2 - 4*(-0.0055)*7))/(2*(-0.0055))
Now we can calculate the value(s) of x.
x = (-1.2 ± sqrt(1.44 + 0.154))/(0.011)
Simplifying further, we have:
x = (-1.2 ± sqrt(1.594))/(0.011)
Calculating the square root, we get:
x ≈ (-1.2 ± 1.262)/0.011
Solving for both cases, we have:
x ≈ (-1.2 + 1.262)/0.011 or x ≈ (-1.2 - 1.262)/0.011
Simplifying, we get:
x ≈ 5.273 or x ≈ -119.091
Since distance cannot be negative, we discard the negative value.
Therefore, when the water is 3 feet above the ocean, it is approximately 5.273 feet from the tugboat.