Question 1174022
Let's break this problem down step-by-step.

**1. Define the Data Points**

* **P(x) Section:**
    * Start: (0, 60)
    * End: (100, 110)
* **Q(x) Section:**
    * Start: (100, 110)
    * End: (150, 70)

**2. Divided-Difference Method for P(x)**

* We need a polynomial that passes through (0, 60) and (100, 110). Since we only have two points, we can use a linear polynomial.

* **Calculate the slope (divided difference):**
    * (110 - 60) / (100 - 0) = 50 / 100 = 0.5

* **Linear Polynomial P(x):**
    * P(x) = 60 + 0.5x

**3. Divided-Difference Method for Q(x)**

* We need a polynomial that passes through (100, 110) and (150, 70). Again, we can use a linear polynomial.

* **Calculate the slope (divided difference):**
    * (70 - 110) / (150 - 100) = -40 / 50 = -0.8

* **Linear Polynomial Q(x):**
    * Q(x) = 110 - 0.8(x - 100)
    * Q(x) = 110 - 0.8x + 80
    * Q(x) = 190 - 0.8x

**4. Polynomial R(x) for the Complete Ride**

* To create a single polynomial R(x) for the entire ride, we would need to use a more complex method, like spline interpolation, to ensure a smooth transition between P(x) and Q(x). However, since P(x) and Q(x) are linear, it will be hard to make a single polynomial.
* Since we have two linear functions that are connected, we can define R(x) as a piecewise function.
    * R(x) = { P(x) if 0 <= x <= 100
             { Q(x) if 100 < x <= 150

**5. Graph Construction**

* **P(x) Graph:**
    * Plot the points (0, 60) and (100, 110).
    * Draw a straight line connecting these points.
* **Q(x) Graph:**
    * Plot the points (100, 110) and (150, 70).
    * Draw a straight line connecting these points.
* **Combined Graph:**
    * Combine the graphs of P(x) and Q(x) on the same coordinate plane.

**Graph Explanation**

* The graph will show two connected line segments.
* The first line segment (P(x)) will rise from (0, 60) to (100, 110).
* The second line segment (Q(x)) will fall from (100, 110) to (150, 70).
* The connection point is (100, 110)

**Summary of Polynomials**

* **(i) Polynomials P(x), Q(x):**
    * P(x) = 0.5x + 60 (for 0 ≤ x ≤ 100)
    * Q(x) = 190 - 0.8x (for 100 < x ≤ 150)
* **(ii) Polynomial R(x):**
    * R(x) = { 0.5x + 60 if 0 <= x <= 100
             { 190 - 0.8x if 100 < x <= 150