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 as such. Thanks! :)

You need to find the four constants, a, b, c, and d.

you have a value for f(x) at x = -1000
write it out:
f(-1000) = -1000 = a (-1000)^3 b(-1000)^2 + c (-1000) + d

you have a value for f(x) at x = +1000
write it out:
f(+1000) = a (1000)^3 + b (1000)^2 + c (1000) + d

you have a value for the slope, f'(x)
at x = -1000
f'(-1000) = 3 a (-1000)^2 + 2 b (-1000) + c

you have a value for the slope at x =+1000
f'(1000) = 3 a (1000)^2 + 2 b (1000) + c

That is four linear equations with four unknowns, a, b, c, and d

Solve those four equations simultaneously and you will have the function and its derivative which you can graph.

The point where the absolute value of the derivative is maximum is the steepest part.

You can find that extreme by setting the derivative of the derivative equal to zero.
6 a x + 2 b = 0
solve for x and calculate the slope at that point.

Sure, I can help you with that! But before we begin, as a clown bot, I must warn you that my explanations might include some humor. So, hold on tight and let's tackle this step by step!

a) To design a section of highway connecting the hillsides, we need to find the values of a, b, c, and d in the function f(x) = ax^3 + bx^2 + cx + d.

First, let's find the slope at points A and B to match the grades of the hillside. The slope of the function at any point is given by the derivative of the function. So, let's find the derivative of f(x).

f'(x) = 3ax^2 + 2bx + c

Now, let's evaluate the slope at A and B.
At A, the slope should be 6%, which is 6/100 = 0.06. The elevation difference at A is 0, so the slope at A should be:

0.06 = 3a(-1000)^2 + 2b(-1000) + c

Simplifying, we have:
3000000a - 2000b + c = 0.06 ...(1)

Similarly, at B, the slope should be 4%, which is 4/100 = 0.04. The elevation difference at B is -50 feet (negative because it's going down), so the slope at B should be:

0.04 = 3a(1000)^2 + 2b(1000) + c - 50

Simplifying, we have:
3000000a + 2000b + c = 0.04 + 50 ...(2)

We now have two equations with three unknowns (a, b, and c). But we have one more condition to satisfy: the elevation difference at the point where the two hillsides meet is 50 feet. Let's incorporate this condition.

Plug in x = 0 into the function f(x) = ax^3 + bx^2 + cx + d and set it equal to 50:

d = 50 ...(3)

So now we have three equations (1, 2, and 3) with three unknowns (a, b, c).

b) Now, let's use a graphing utility to graph the model. We will use the function f(x) = ax^3 + bx^2 + cx + d with the values of a, b, c, and d that we found.

c) Next, let's graph the derivative of the model. We found the derivative to be f'(x) = 3ax^2 + 2bx + c. We can plot this function on the same graph.

d) To determine the grade at the steepest part of the transitional section of the highway, we need to find the maximum or minimum value of the derivative function. This can be done by finding the critical points, where the derivative is equal to zero or undefined.

So, we set f'(x) = 0 and solve for x to find the critical points.

3ax^2 + 2bx + c = 0

Now, we have an equation with two unknowns (a and b). Solving this equation will give us the x-coordinate of the steepest part.

Once we determine the value of x, we can plug it into the derivative function to find the grade (slope) at the steepest part.

And that's it! By following these steps, you can design a section of the highway connecting the hillsides and determine the grade at the steepest part. I hope this helps, and remember, if the math gets tough, just sprinkle some laughs along the way!

To design a section of highway connecting the hillsides, we need to find the values of a, b, c, and d in the function f(x) = ax^3 + bx^2 + cx + d that satisfy the given conditions.

Step 1: Setting up the equations
a) To determine the values of a, b, c, and d, we need to set up a system of equations using the given conditions.

At the point where the two hillsides come together:
f(-1000) = -6
f(1000) = 4

Slope of the model matches the grade of the hillside at points A and B:
f'(-1000) = -6/100
f'(1000) = 4/100

Step 2: Solving the system of equations
a) Substitute the x-values into the function to get two equations:
-1000^3a + 1000^2b - 1000c + d = -6
1000^3a + 1000^2b + 1000c + d = 4

b) Substitute the x-values into the derivative of the function to get two equations:
3*(-1000)^2a + 2*1000b - 6c = -6/100
3*1000^2a + 2*1000b + 6c = 4/100

Step 3: Solving the system of equations
a) Convert the equations into a matrix form:
[ (-1000^3) (1000^2) (-1000) (1) | -6 ]
[ (1000^3) (1000^2) 1000 (1) | 4 ]

[ 3*(-1000)^2 2*1000 -6 | -6/100 ]
[ 3*1000^2 2*1000 6 | 4/100 ]

b) Use matrix operations or a calculator to solve the system of equations to find the values of a, b, c, and d.

Step 4: Graphing the model
a) Using a graphing utility, substitute the found values of a, b, c, and d into the function f(x) = ax^3 + bx^2 + cx + d to graph the model.

Step 5: Graphing the derivative
a) Using a graphing utility, substitute the found values of a, b, and c into the derivative function f'(x) = 3ax^2 + 2bx + c to graph the derivative.

Step 6: Determine the grade at the steepest part of the transitional section of the highway
a) The grade corresponds to the slope of the function at that point. Look for the highest point or peak in the graph of the derivative function to determine the steepest part of the transitional section of the highway.

Note: The step-by-step instructions provide a general guideline to solve the problem. However, specific details and calculations may vary depending on the graphing utility or calculator used to solve the equations and graph the functions.

To design a section of highway connecting the hillsides, we need to determine the coefficients a, b, c, and d in the function f(x) = ax^3 + bx^2 + cx + d based on the given conditions.

a) First, let's consider the slope at points A and B, which must match the grades of the hillside.

At point A:
The grade of the hillside is 6%, which means for every 100 units of horizontal distance, there is a rise of 6 units. We can represent this as a slope of 6/100. Since point A is 1000 feet to the left of the transitional section, the slope of the function at point A should also be 6/100.

So, the derivative of the function at x = -1000 should be 6/100.

f'(x) = 3ax^2 + 2bx + c
f'(-1000) = 3a(-1000)^2 + 2b(-1000) + c = 6/100

At point B:
The grade of the hillside is 4%, which means for every 100 units of horizontal distance, there is a rise of 4 units. We can represent this as a slope of 4/100. Since point B is 1000 feet to the right of the transitional section, the slope of the function at point B should also be 4/100.

So, the derivative of the function at x = 1000 should be 4/100.

f'(x) = 3ax^2 + 2bx + c
f'(1000) = 3a(1000)^2 + 2b(1000) + c = 4/100

Now, we have two equations with three unknowns (a, b, and c). To solve this system of equations, we need one more equation.

At the point where the two hillsides come together, there is a 50-foot difference in elevation. This means the function values at x = -1000 and x = 1000 should differ by 50.

So, plugging in the values for x = -1000 and x = 1000 into the function f(x), we get:

f(-1000) = a(-1000)^3 + b(-1000)^2 + c(-1000) + d = 50
f(1000) = a(1000)^3 + b(1000)^2 + c(1000) + d = 0

Now we have a system of four equations with four unknowns:
3a(-1000)^2 + 2b(-1000) + c = 6/100
3a(1000)^2 + 2b(1000) + c = 4/100
a(-1000)^3 + b(-1000)^2 + c(-1000) + d = 50
a(1000)^3 + b(1000)^2 + c(1000) + d = 0

Next, we can use a graphing utility to graph the model and its derivative:

b) Using a graphing utility, plot the function f(x) = ax^3 + bx^2 + cx + d. Adjust the values of a, b, c, and d to get the desired shape of the graph based on the given conditions. This will give you a visual representation of the highway design connecting the hillsides.

c) To graph the derivative of the model, plot the function f'(x) = 3ax^2 + 2bx + c using the same graphing utility. Again, adjust the values of a, b, and c to match the grade of the hillside at points A and B.

d) To determine the grade at the steepest part of the transitional section of the highway, examine the graph of the derivative. The steepest part of the graph corresponds to the highest value of the derivative. Find the x-coordinate(s) where the derivative is the highest and calculate the corresponding slope at that point.

By following these steps, you should be able to design a section of highway connecting the hillsides, graph the model and its derivative, and determine the grade at the steepest part of the transitional section of the highway.