A section of highway connecting two hillsides with grades of 6% and 4% is to be build between two points that are separated by a horizontal distance of 2000 feet. At the point where the two hillsides come together, there is a 50-foot difference in elevation.
a) Design a section of highway connecting the hillsides modeled by the function f(x) = ax^3 + bx^2 + cx + d (-1000 less than or equal to x less than or equal to 1000). At the points A and B, the slope of the model must match the grade of the hillside.
b) Use a graphing utility to graph the model.
c) Use a graphing utility to graph the derivative of the model.
d) Determine the grade at the steepest part of the transitional section of the highway.
I need to show work step-by-step for this, so please format your answer like that. Thanks! :)
I started by putting point A at the origin (0,0), then point B would be at (2000,50)
I realized I was going to work with very large numbers, so I "scaled" my graph back by a factor of 50:1
so point A was still (0,0) but point B became (40,1)
let f(x) = ax^3 + bx^2 + cx + d
at (0,0) this would give me d = 0
so f(x) = ax^3 + bx^2 + cx
I then subst. (40,1) into that to get
1 = 64000a + 1600b + 40c (equ#1)
also f'(x) = 3ax^2 + 2bx + c
we know at (0,0) slope = .04
so ....c=.04
we know at (40,1) slope = .06
so.. .06 = 4800a + 80b + c
but c=.04
4800a + 80b = .02 (equ#2)
I then put c=.04 into equ#1, and solved this with equ#2 to get
a=1/32000
b=-13/8000
c=1/25
then finally
f(x) = (1/32000)x^3 - (13/8000)x^2 + (1/25)x
I set the derivative of that equal to zero, there was no real solution, so there is no max/min to this function.
there is a point of inflection at x=17.3
which translates into 17.3*50 = 865 m horizontal from A
My guess is that the slope of .06 is the largest in your domain
Please check my work, hard to do arithmetic like this while watching a football game, lol
To design the section of highway connecting the hillsides, we will start by finding the values of a, b, c, and d in the function f(x) = ax^3 + bx^2 + cx + d.
Step 1: Define the points A and B on the highway section.
Let point A be a horizontal distance (x-coordinate) of -1000 feet and point B be a horizontal distance of 1000 feet.
Step 2: Set up equations for points A and B.
At point A, the slope of the model must match the grade of the hillside, which is 6% or 0.06. The slope of the function f(x) at point A is given by the derivative f'(x).
At point B, the slope of the model must match the grade of the hillside, which is 4% or 0.04. The slope of the function f(x) at point B is given by the derivative f'(x).
Step 3: Calculate the slope at points A and B.
Using the derivative, we can determine the slope of the function f(x) at any point. Let's find the slope at points A and B.
f'(x) = 3ax^2 + 2bx + c
At point A (-1000 feet), the slope should be 0.06:
f'(-1000) = 0.06
0.06 = 3a(-1000)^2 + 2b(-1000) + c [Substitute x = -1000 in f'(x)]
At point B (1000 feet), the slope should be 0.04:
f'(1000) = 0.04
0.04 = 3a(1000)^2 + 2b(1000) + c [Substitute x = 1000 in f'(x)]
Step 4: Solve the system of equations.
Solve the system of equations formed in step 3 to find the values of a, b, and c.
0.06 = 3a(-1000)^2 + 2b(-1000) + c
0.04 = 3a(1000)^2 + 2b(1000) + c
Solve the above system of equations simultaneously to determine the values of a, b, and c.
Step 5: Determine the value of d.
Since the difference in elevation between the hillsides is 50 feet, we can obtain the value of d by substituting the coordinates of either point A or B into f(x). Let's use the coordinates of point A:
f(-1000) = a(-1000)^3 + b(-1000)^2 + c(-1000) + d
50 = a(-1000)^3 + b(-1000)^2 + c(-1000) + d
Now, solve the equation above for d.
Step 6: Create the function f(x).
Now that we have determined the values of a, b, c, and d, we can construct the function f(x) = ax^3 + bx^2 + cx + d:
f(x) = ax^3 + bx^2 + cx + d
Substitute the values of a, b, c, and d obtained in the previous steps into the function.
a) Design a section of the highway connecting the hillsides modeled by the function f(x) = ax^3 + bx^2 + cx + d (-1000 ≤ x ≤ 1000).
b) To graph the model, use a graphing utility and plot the function f(x) with the determined values of a, b, c, and d.
c) To graph the derivative of the model, use a graphing utility and plot the function f'(x) = 3ax^2 + 2bx + c with the determined values of a, b, and c.
d) To determine the grade at the steepest part of the transitional section of the highway, look for the highest or lowest value of the derivative f'(x). The grade corresponds to the slope of the hillside at that point.