The problem is:
In order to build a highway, it is necessary to fill in a section of a valley where the grades (slopes) of the sides are 9% amd 6%. 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 distance between points A and B are 1000 feet.
a) Find a quadratic function y=ax^2 +bx +c, -500<_ x<_ 500, that describes the top of the filled region.
b) Construct a table giving the depths of d of the fill for x= -500, -400, -300, -200, -100, 0, 100, 200, 300, 400, 500
c) What wil be the lowest point on the completed highway? Will it be directly over the point where the two hillsides come together?
I need major explanation of how to do each part. I'm extremely confused. And calculus obviously has to be used to solve it. It is modelization by the way. Please help! Thank you!
In order to build a highway, it is necessary to fill in a section of a valley where the grades (slopes) of the sides are 9% amd 6%. 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 distance between points A and B are 1000 feet.
a) Find a quadratic function y=ax^2 +bx +c, -500<_ x<_ 500, that describes the top of the filled region.
b) Construct a table giving the depths of d of the fill for x= -500, -400, -300, -200, -100, 0, 100, 200, 300, 400, 500
c) What wil be the lowest point on the completed highway? Will it be directly over the point where the two hillsides come together?
I need major explanation of how to do each part. I'm extremely confused. And calculus obviously has to be used to solve it. It is modelization by the way. Please help! Thank you!
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(a) The first part of this question has been asked and solved previously by me, here: http://answers.yahoo.com/question/index;…
Briefly, a = 75/1,000,000, b = 15/1,000, c = 35,625/1,000
(b) The value of depth here is exactly the value of y. So construct a table showing the given values of x and their corresponding values y.
(c) To compute the lowest point, dy/dx = 0, i.e., 2ax + b = 0 => x = -b/2a = 1,000,000 * 15 / (2 * 75 * 1000) = 1000/10 = 100. Since the solution of (a) assumes that the bottom of the valley is at the origin, it follows that this lowest point is 100 ft displaced (towards the 6% gradient) from the point where the hills come together.
[Edit] I did use derivatives in my solution to (a). The step where I equate the slope of the tangent to 2ax + b, is where the derivative is used.
[Edit] To address Matt's comment: I did not assume that the parabola has constant slope but that the tangent has constant slope. Your solution c = 0 is incorrect. You say that you have 3 equations in three unknowns; you actually have 5 unknowns including y1 and y2. You have not used the fact that the two points where the mountains are tangent to the parabola also *lie on the parabola*. In fact, this system of equations correctly has 3 unknowns (a,b,c since y1,y2 can be calculated using the slopes of the lines, as I did), and 4 equations (2 from the tangent slopes and 2 from the intersection of the parabola and the lines). So it is overconstrained. It just turns out that in this case there is a solution.
Briefly, a = 75/1,000,000, b = 15/1,000, c = 35,625/1,000
(b) The value of depth here is exactly the value of y. So construct a table showing the given values of x and their corresponding values y.
(c) To compute the lowest point, dy/dx = 0, i.e., 2ax + b = 0 => x = -b/2a = 1,000,000 * 15 / (2 * 75 * 1000) = 1000/10 = 100. Since the solution of (a) assumes that the bottom of the valley is at the origin, it follows that this lowest point is 100 ft displaced (towards the 6% gradient) from the point where the hills come together.
[Edit] I did use derivatives in my solution to (a). The step where I equate the slope of the tangent to 2ax + b, is where the derivative is used.
[Edit] To address Matt's comment: I did not assume that the parabola has constant slope but that the tangent has constant slope. Your solution c = 0 is incorrect. You say that you have 3 equations in three unknowns; you actually have 5 unknowns including y1 and y2. You have not used the fact that the two points where the mountains are tangent to the parabola also *lie on the parabola*. In fact, this system of equations correctly has 3 unknowns (a,b,c since y1,y2 can be calculated using the slopes of the lines, as I did), and 4 equations (2 from the tangent slopes and 2 from the intersection of the parabola and the lines). So it is overconstrained. It just turns out that in this case there is a solution.