A bridge over a river is supported by an arch. The function that describes the arch us h(x) = -0.0274x^2 + 1.34033x, where h(x) is the height, in metres, of the arch above the water at any distance, x, in metres, from one end of the bridge. Could a sail boat with an 18m mast sail under the bridge.
Convert equation to its canonical form by completing squares:
h(x) = -0.0274x^2 + 1.34033x
=-0.0274(x-24.46)²+16.3913
which means that the vertex is located at (24.46,16.3913).
Determine if the vessel can goes through.
Did it go through?
To determine whether a sailboat with an 18m mast can sail under the bridge, we need to find the height of the arch at its highest point. If the height of the arch is greater than or equal to 18m at any distance along the bridge, then the sailboat cannot pass under the bridge.
The given function h(x) = -0.0274x^2 + 1.34033x represents the height of the arch above the water at any distance x from one end of the bridge.
To find the maximum height of the arch, we need to find the vertex of the quadratic function. The vertex form of a quadratic function is given by h(x) = a(x - h)^2 + k, where (h, k) represents the vertex.
In our case, the quadratic function is h(x) = -0.0274x^2 + 1.34033x. Converting it to vertex form, we have h(x) = -0.0274(x - 24.46)^2 + k.
Comparing this equation to the vertex form, we can see that h = 24.46 and k is the maximum height of the arch.
Therefore, the highest point of the arch is at a height of k meters above the water.
To calculate the maximum height, we can set the derivative of the function equal to zero (since the vertex occurs at the maximum point). The derivative of h(x) = -0.0274x^2 + 1.34033x is given by h'(x) = -0.0548x + 1.34033.
Setting h'(x) = 0 and solving for x, we get:
-0.0548x + 1.34033 = 0
-0.0548x = -1.34033
x = -1.34033 / -0.0548
x ≈ 24.46
Substituting this x-value back into the original function, we can find the maximum height:
h(24.46) = -0.0274(24.46)^2 + 1.34033(24.46)
h(24.46) ≈ 18.09
The maximum height of the arch is approximately 18.09 meters.
Since the maximum height is below 18 meters, a sailboat with an 18m mast can sail under the bridge.