Question

In order to build a highway, it is necessary to fill a section of a valley where the grades (slopes) of the sides are 9% and 6% (see figure). The top of the filled region will have the shape of a parabolic arc that is tangent to the two slopes at the points A and B. The horizontal distances from A to the y-axis and from B to the y-axis are both 500 feet.

Answer 1

(a.) Find the coordinates of A and B.
(b.) Find a quadratic function y = ax2 + bx + c for -500 <= x <=500 that describes the top of the filled region.
(c.) Construct a table giving the depths d of the fill for x = -500, -400, -300, -200, -100, 0, 100, 200, 300, 400 and 500
(d.) What will be the lowest point on the completed highways?
Will it be directly over the point where the two hillsides come together?

Answer 2

I can help you set up the integrals and find all the information you need. However, I do not understand the percentages associated with the slopes. What's the 9% for?

Answer 3

the grade

Answer 4

I magine that is enginerese, I do not speak it. In plain english, what does it represent?

Answer 5

I know the Tangent line goes off of the grade

Answer 6

It is much like a slope

Answer 7

Extrema on a interval

Answer 8

grade = slope

Answer 9

I got it now. Let me write it up.

Answer 10

okay thanks

Answer 11

From Wikipedia I gather Civil Engineers measure slopes in percentages. So, here we go;

The slope is measure from the base, or horizontal at ground level. So, at \(9\%\) going \(500 ft\) to the left that makes \(500*\frac{9}{100}=5*9=45ft\) so that's the height to the left.

For the right we do the same with \(6\%\) this time. Once you do, you will get \(30ft\).


So Part (a) is done. I think you can figure out how to put that together to find the points \(A\) and \(B\).

Next part is coming up.

Answer 12

Since the origin is in our function by construction \(c=0\). By construction I mean that the problem is set up so the origin is the bottom of the valley, the point on the horizontal at the base where we measure the grades from.

So we have \(ax^2+bx=0\) plugging in our points gets us a two equations in two variables. You can do the substitution method, the addition method or make a matrix.

Answer 13

So far so good?

Answer 14

yes thanks

Answer 15

I still need help with c. and d.