Road Design Roads are often designed with parabolic surfaces...

Road Design Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than it is on the sides (see figure). (a) Find an equation of the parabola that models the road surface. (Assume that the origin is at the center of the road.) (b) How far from the center of the road is the road surface 0.1 foot lower than in the middle?

Road Design Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than it is on the sides (see figure).
(a) Find an equation of the parabola that models the road surface. (Assume that the origin is at the center of the road.)
(b) How far from the center of the road is the road surface 0.1 foot lower than in the middle?

Key Concepts

Symmetry of Parabolas

Parabolas are symmetric about a vertical line passing through the vertex. This symmetry means that the distance from the vertex to any point on one side of the parabola will have a corresponding point on the opposite side with the same vertical value. This property simplifies computations related to distances from the vertex, such as determining horizontal offsets for given vertical differences.

Determining the Coefficient a

When modeling a parabolic surface, once you have identified the vertex, you can determine the coefficient a by substituting another known point on the curve into the vertex form. This coefficient controls the steepness and the direction (upward or downward opening) of the parabola, ensuring that the model accurately represents the physical scenario.

Solving Quadratic Equations

Often, it is necessary to find the x-values for which the quadratic function attains a specific y-value. This requires setting the quadratic function equal to the given value and solving the resulting equation, typically using algebraic manipulation or the quadratic formula. Such techniques are essential when determining distances along the curve where the function reaches a certain height.

Quadratic Functions

Quadratic functions, expressed in the standard form y = ax² + bx + c, model parabolic shapes. They are widely used to represent physical curves such as roads because they have a constant second derivative and provide a smooth curve that can be easily manipulated by adjusting the parameter a to control the curvature.

Vertex Form of a Parabola

The vertex form of a quadratic function, written as y = a(x - h)² + k, explicitly shows the vertex (h, k) of the parabola. This form is particularly useful when the maximum or minimum point of the parabola is known, as in many design applications where the highest or lowest point is fixed, and the rest of the curve is defined relative to it.

Transcript

00:01 So in this problem, we are kind of given a diagram about how roads are designed.

00:04 We're told that the length is 32 feet and that its height is 0 .4 feet.

00:11 So in part a, what they want us to do is find an equation for the parabola that will model this road surface.

00:18 And what they want us to do is assume that the origin is at the center of the road.

00:22 So if the origin is at the center of the road, that would be right here.

00:26 So this would be our y -axis.

00:28 Let me draw that a little straight there.

00:30 So this would be our y -axis, which means this right here would be our x -axis.

00:35 Therefore, because the center is at our origin, we know that the vertex for this parabola will be at the ordered pair, 0 -0.

00:44 So now what we're going to need to do is we're going to need to find another ordered pair.

00:49 Well, keep in mind that the total horizontal distance is 32 feet.

00:52 Because that's at the center, we can find the distance between the center and this furtiveous point on the right.

00:58 That would be half of it, which is 16 feet.

01:00 So the x coordinate of this point would be 16.

01:04 And because we're told the height of the road is 0 .4, but in this case it's going down, that means the y coordinate at when x is 16 would be negative 0 .4.

01:15 So now that we know the vertex and a point on our graph, we now can go ahead and find our focus.

01:23 So we're going to use our general equation, x minus h squared, is equal to 4 times p times y, minus k.

01:34 So for x and y, we're going to substitute in our ordered pair, 16 and negative 0 .4.

01:40 And then for h and k, we substitute the x and y value of our vertex.

01:44 So when we do this, we're going to get 16 minus 0 squared equals 4 p times negative 0 .4 minus 0...

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