The height above water level of a curved arch for a bridge can be modeled by h(x) = -0.009x^2 + 1.08x − 0.4, where x is the distancein feet from the point where the left leg of the arch is planted in the ground.

A boat traveling under the arch is 26.5 feet tall. What is the vertical distance between the tallest point on the boat and the tallest point on the arch as the boat passes under the bridge?

~I got 33.5 doing the problem and i got it wrong can you please explain it to me and help me

where is the vertex of that parabola?

You do not say if you know calculus so I will assume we do it the hard way with algebra

.009 x^2 -1.08 x = -h -.4
x^2 - 120 x = -111h - 44.4
x^2 -120 x + 3600 = -111h + 3556
(x-36)^2 = -111(h-32)
vertex at x = 36 and h = 32
32 -26.5 = 5.5

Thank's for helping me.

You are welcome.

Sure! To find the vertical distance between the tallest point on the boat and the tallest point on the arch, we need to find the difference between the height of the boat (26.5 feet) and the height of the arch at that particular point.

The equation given to model the height of the arch is h(x) = -0.009x^2 + 1.08x - 0.4, where x is the distance in feet from the point where the left leg of the arch is planted in the ground.

Let's denote the distance from the left leg of the arch to the boat as d. We can set up an equation to find d by equating the height of the arch at distance d to the height of the boat:

-0.009d^2 + 1.08d - 0.4 = 26.5

Now we need to solve this equation to find the value of d.

First, let's rearrange the equation to bring all terms to one side:

-0.009d^2 + 1.08d - 0.4 - 26.5 = 0

Simplifying, we get:

-0.009d^2 + 1.08d - 26.9 = 0

Next, we can apply the quadratic formula: d = (-b ± √(b^2 - 4ac)) / 2a

For our equation, a = -0.009, b = 1.08, and c = -26.9. Substituting these values into the quadratic formula, we have:

d = (-1.08 ± √(1.08^2 - 4(-0.009)(-26.9))) / (2(-0.009))

Calculating the expression inside the square root, we have:

√(1.1664 + 0.96792) = √2.13432 ≈ 1.46

Substituting this value into our equation, we now have:

d = (-1.08 ± 1.46) / (-0.018)

Now we can solve for the two possible values of d:

d1 = (-1.08 + 1.46) / (-0.018) ≈ -61.11
d2 = (-1.08 - 1.46) / (-0.018) ≈ 84.44

Since distance cannot be negative in this context, we discard d1 and keep d2.

Therefore, the distance from the left leg of the arch to the boat is approximately 84.44 feet.

To find the vertical distance between the tallest points on the boat and the arch, we can substitute this value of d into the equation for the height of the arch:

h(84.44) = -0.009(84.44)^2 + 1.08(84.44) - 0.4 ≈ 33.473

Therefore, the vertical distance between the tallest point on the boat and the tallest point on the arch as the boat passes under the bridge is approximately 33.473 feet.