if the parkway is 60 feet wide, the minimum clearance is 15 feet and the parabolic arch is 25 feet high, how long is the arch?
If the clearance is 15 ft at the edges of the pavement, then we have
y = 25-ax^2
y(30) = 15
so now we know that
25 - 900a = 15
a = 1/90
now, knowing that
y = 25 - 1/90 x^2, we have
y=0 at x = 15√10 ≈ 47.43
so the arch is 94.86 ft wide at its base
I assume you are not really after the actual length of the curve, since that requires calculus.
Well, if the arch is already 25 feet high, I don't think we need to worry about its length. It's not like the length of our arch is trying to compensate for anything in particular, like an arch mid-life crisis or something. So, we can just enjoy its height without questioning its length.
To find the length of the arch, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the width of the parkway is the base of the triangle, the height of the arch is one of the legs, and the length of the arch (which is what we are trying to find) is the hypotenuse.
Let's call the length of the arch "x". We can set up the equation as follows:
x² = 60² + 25²
x² = 3600 + 625
x² = 4225
Taking the square root of both sides:
x = √4225
x = 65
Therefore, the length of the arch is 65 feet.
To determine the length of the arch, we can use the information provided about the width, minimum clearance, and height of the arch.
Given that the width of the parkway is 60 feet, the clearance is 15 feet, and the arch is 25 feet high, we can draw a triangle to represent the arch.
Let's label the length of the arch as "x".
The triangle can be divided into two parts: a right-angled triangle and an isosceles triangle.
In the right-angled triangle, the height of the arch (25 feet) is the opposite side, the clearance (15 feet) is the adjacent side, and the length of the arch (x) is the hypotenuse.
Using the Pythagorean theorem, we can calculate the length of the arch:
x^2 = (25^2) + (15^2)
x^2 = 625 + 225
x^2 = 850
x ≈ √850
x ≈ 29.15 feet
Therefore, the length of the arch is approximately 29.15 feet.