a parkway 80 ft. wide is spanned by a parabolic arc 100 ft. long along the horizontal, if the parkway is in the center, how high must the vertex of the arch be in order to give a minimum clearance of 20 over the parkway?
To find the height of the vertex of the arch that gives a minimum clearance of 20 feet over the parkway, we can use a basic mathematical concept called "optimization."
Here's how we can solve the problem step by step:
Step 1: Visualize the problem
Draw a diagram of the parkway with the parabolic arc spanning across it. The width of the parkway is 80 feet, and the length of the arc (the horizontal distance) is 100 feet. We want to find the height of the vertex of the parabolic arc.
Step 2: Set up the problem
Let's assume that the vertex of the parabolic arc is (0, h), where h represents the height we're trying to find. Since the parkway is in the center, the width on each side of the vertex is half of the total width, which is 80/2 = 40 feet.
Step 3: Set up the equation of the parabolic arc
The equation of a parabola in vertex form is y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. In this case, the vertex is (0, h). Since the vertex is at the center of the arch, the equation becomes y = ax^2 + h.
Step 4: Find the equation of the parabolic arc
To find the equation of the parabola, we need to find the value of a. Since the parabolic arc spans 100 feet horizontally across the parkway, the length of the arc is equal to the integral of the function.
∫[-40, 40] √(1 + (dy/dx)^2) dx = 100, where dy/dx is the derivative of y with respect to x.
Simplifying this equation leads to ∫[-40, 40] √(1 + 4a^2x^2) dx = 100.
Step 5: Solve the integral equation
Evaluate the integral and solve for a:
2/3 * (40√(1 + 1600a^2) + sinh^(-1)(40√(1 + 1600a^2)) - sinh^(-1)(-40√(1 + 1600a^2))) = 100
Step 6: Solve for h
Once you find the value of a, you can substitute it back into the equation y = ax^2 + h, and set y + 20 = ax^2 + h (to account for the minimum clearance of 20 feet). Then, solve for h.
Step 7: Find the minimum clearance
Now that you know the height of the vertex (h), you can calculate the minimum clearance by subtracting 20 feet from the height.
That's it! By following these steps, you can find the height of the vertex of the arch that gives a minimum clearance of 20 feet over the parkway.