Te arch of the Sydeny Harbor Bridge can be modeled by y= -0.00211x^2 + 1.06x where x is the distance (in meters) from the left pylons and y is the height (in meters) of the arch above the water. For what distances x is the arch above the road? (It has a picture and in it the distance from the bottom of the pylons to the road is 52 m)
To determine for what distances x the arch is above the road, we need to find the values of x when the height of the arch (y) is greater than the height of the road (52 meters).
The equation of the arch is given as y = -0.00211x^2 + 1.06x.
To find the values of x when y is above 52 meters, we set up the inequality:
-0.00211x^2 + 1.06x > 52
Now let's solve this inequality step by step:
Step 1: Subtract 52 from both sides:
-0.00211x^2 + 1.06x - 52 > 0
Step 2: Rearrange the equation in descending order of the powers of x:
-0.00211x^2 + 1.06x - 52 = 0
Step 3: To solve this quadratic equation, we can use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solution for x is given by:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation -0.00211x^2 + 1.06x - 52 = 0, the values of a, b, and c are as follows:
a = -0.00211
b = 1.06
c = -52
Substituting these values into the quadratic formula, we get:
x = (-1.06 ± sqrt(1.06^2 - 4 * -0.00211 * -52)) / (2 * -0.00211)
Simplifying this equation gives us two possible values for x. These values will indicate the distances from the left pylons for which the arch is above the road.
Please note that there will be two distances because the graph of the quadratic equation y = -0.00211x^2 + 1.06x is a downward-opening parabola, so it intersects the horizontal axis (y = 52) at two points.