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Hmmm this problem sounds like we need to use our calculus knowledge to solve an algebra problem. Since algebra is always the hard part, it is admittedly going to be a hard question if care is not taken.
9% and 6% are the grades of the valley so they are slope values. so dy/dx(A) = 9% and dy/dx(B) = 6%, where A and B are actually the x-values of the given points A and B. So if I were to doodle a little picture I would encounter our first assumption that we need to make pretty quick. One of these should be negative or else our valley is always sloping in the same direction. So lets make life VERY easy for ourselves and take advantage of the fact we know the points are 1000 ft apart and construct our own coordinate system.
Let point A = (-500, y1)
Let point B = (+500,y2)
Let dy/dx(-500) = -0.09
Let dy/dx(+500) = 0.06
So right there we established that at our negative portion the curve is sloping down, and at our positive portion the curve is sloping up.
eq1) y = ax^2 + bx + c
eq2) dy/dx(-500) = -0.09
eq3) dy/dx(500) = 0.06
We have 3 unknowns (a, b, c) and we have 3 equations, nice.
dy/dx = 2ax + b, since it is easy enough to differentiate eq1.
dy/dx(-500) = 2a(-500) + b = -0.09
b -1000a = -0.09
dy/dx(+500) = 2a(+500) + b = +0.06
b + 1000a = 0.06
Now we can at least solve for a and b (using whatever your method of choice is).
(I just plugged these into a matrix on my calculator out of laziness)
a = 0.000075
b = -0.015
y = 0.000075 x^2 - 0.015x + c
Great, we don't know c.... Well since we are creating our own coordinate system, we would obviously want to go through the point (0,0), right?
0 = 0.000075(0)^2 - 0.015(0) + c
Ah ok, c = 0, good.
y = 0.000075 x^2 - 0.015x
From here on out we simply just plug in our values to fill in the tables for part b.
A word of caution, and you should know this already from just the steps we took to get this far, our vertex is probably (and in fact does not) is not going to occur at the origin. But that was never part of the question, so don't flip out, we just had to make logical assumptions to solve the problem.
9% and 6% are the grades of the valley so they are slope values. so dy/dx(A) = 9% and dy/dx(B) = 6%, where A and B are actually the x-values of the given points A and B. So if I were to doodle a little picture I would encounter our first assumption that we need to make pretty quick. One of these should be negative or else our valley is always sloping in the same direction. So lets make life VERY easy for ourselves and take advantage of the fact we know the points are 1000 ft apart and construct our own coordinate system.
Let point A = (-500, y1)
Let point B = (+500,y2)
Let dy/dx(-500) = -0.09
Let dy/dx(+500) = 0.06
So right there we established that at our negative portion the curve is sloping down, and at our positive portion the curve is sloping up.
eq1) y = ax^2 + bx + c
eq2) dy/dx(-500) = -0.09
eq3) dy/dx(500) = 0.06
We have 3 unknowns (a, b, c) and we have 3 equations, nice.
dy/dx = 2ax + b, since it is easy enough to differentiate eq1.
dy/dx(-500) = 2a(-500) + b = -0.09
b -1000a = -0.09
dy/dx(+500) = 2a(+500) + b = +0.06
b + 1000a = 0.06
Now we can at least solve for a and b (using whatever your method of choice is).
(I just plugged these into a matrix on my calculator out of laziness)
a = 0.000075
b = -0.015
y = 0.000075 x^2 - 0.015x + c
Great, we don't know c.... Well since we are creating our own coordinate system, we would obviously want to go through the point (0,0), right?
0 = 0.000075(0)^2 - 0.015(0) + c
Ah ok, c = 0, good.
y = 0.000075 x^2 - 0.015x
From here on out we simply just plug in our values to fill in the tables for part b.
A word of caution, and you should know this already from just the steps we took to get this far, our vertex is probably (and in fact does not) is not going to occur at the origin. But that was never part of the question, so don't flip out, we just had to make logical assumptions to solve the problem.
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