The arch of the Gateshead Millennium Bridge forms a parabola with equation y=-0.016(x-52.5)^2+45 where x is the horizontal distance(in meters) from the arch's left end and y is the distance (in meters)from the base of the arch. What is the width of the arch?

I got 105 meters by taking p=52.5 and adding q=52.5. Is 105 meters the width?

105

To find the width of the arch, we need to find the distance between the two points where the parabola intersects the x-axis (y = 0).

Step 1: Set y = 0 in the equation:
0 = -0.016(x - 52.5)^2 + 45

Step 2: Solve for x:
-0.016(x - 52.5)^2 = -45
(x - 52.5)^2 = -45 / -0.016
(x - 52.5)^2 = 2812.5
x - 52.5 = ±√(2812.5)
x - 52.5 = ±53.02

Step 3: Solve for x:
x = 52.5 ± 53.02

The two possible x-values are:
x1 = 52.5 + 53.02 = 105.52
x2 = 52.5 - 53.02 ≈ -0.52

Step 4: Calculate the width of the arch:
Width = |x2 - x1|
Width = |-0.52 - 105.52|
Width ≈ 106.04 meters

Therefore, the width of the arch is approximately 106.04 meters.

To find the width of the arch, we first need to understand what the term "width" refers to in the context of a parabola.

In this equation, y represents the distance from the base of the arch, while x represents the horizontal distance from the left end of the arch. To find the width, we need to determine the horizontal distance from one end of the arch to the other.

The vertex form of a parabola equation is given by y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.

Comparing this with the given equation, y = -0.016(x - 52.5)^2 + 45, we can see that the vertex is located at (52.5, 45).

Since the parabola is symmetric, the width of the arch can be determined by finding the distance between the vertex and any one of the x-intercepts (the points where the parabola intersects the x-axis).

To find the x-intercepts, we set y = 0 in the equation:

0 = -0.016(x - 52.5)^2 + 45

Rearranging and isolating the variable, we get:

0.016(x - 52.5)^2 = 45

Now we divide both sides by 0.016:

(x - 52.5)^2 = 45 / 0.016

(x - 52.5)^2 = 2812.5

Taking the square root of both sides:

x - 52.5 = ±√2812.5

Adding 52.5 to both sides, we have:

x = 52.5 ± √2812.5

Therefore, the x-intercepts are given by x = 52.5 + √2812.5 and x = 52.5 - √2812.5.

To find the width, we subtract the smaller x-intercept from the larger one:

Width = (52.5 + √2812.5) - (52.5 - √2812.5)

Simplifying further, we get:

Width = 2√2812.5

Using a calculator, we can evaluate the expression:

Width ≈ 105.74 meters

Therefore, the width of the arch is approximately 105.74 meters.