The shape of a supporting arch can be modeled by h(x)= -0.03^2+3x, where h(x) represents the height of the arch and x represents the horizontal distance from one end of the base of the arch in meters. Find the maximum height of the arch.

You probably meant h(x)= -0.03x^2+3x

several ways to do this
1. by Calculus, find derivative, set it equal to zero and solve

2. by completing the square and changing the equation to the standard form of a parabola

3. in this case this is the easiest way:
since it factors to

h(x)= x(-0.03x+3)

and when h(x) = 0, x = 0 or -.03x+3=0
yielding x = 100

so the x-intercept of this parabola are 0 and 100, which means that the vertex is at the midway, or at x=50

so h(50) = 75

To find the maximum height of the arch, we need to determine the vertex of the quadratic equation h(x) = -0.03x^2 + 3x.

The vertex of a quadratic equation in the form of h(x) = ax^2 + bx + c is given by the formula x = -b / (2a).

In this case, a = -0.03 and b = 3. Plugging these values into the formula, we get:

x = -3 / (2 * -0.03)
x = -3 / -0.06
x = 50

Therefore, the maximum height of the arch occurs at x = 50 meters.

To find the corresponding height h(x), we substitute this value of x into the equation h(x) = -0.03x^2 + 3x:

h(50) = -0.03(50)^2 + 3(50)
h(50) = -0.03(2500) + 150
h(50) = -75 + 150
h(50) = 75

Hence, the maximum height of the arch is 75 meters.

To find the maximum height of the arch, we need to find the highest point on the graph of the function h(x).

Step 1: Recall that the maximum height occurs at the vertex of the quadratic function. The vertex of a quadratic function in the form h(x) = ax^2 + bx + c can be found using the formula: x = -b / (2a)

Step 2: In our case, the function h(x) = -0.03x^2 + 3x is already in the form ax^2 + bx, where a = -0.03 and b = 3. We can directly apply the formula to find the x-coordinate of the vertex.

x = -3 / (2 * (-0.03)) = 50

Step 3: Now that we know the x-coordinate of the vertex, we can find the maximum height by substituting this x-value back into the function h(x).

h(x) = -0.03x^2 + 3x

h(50) = -0.03(50)^2 + 3(50)

h(50) = -0.03(2500) + 150

h(50) = -75 + 150

h(50) = 75

Therefore, the maximum height of the arch is 75 meters.