Question 1174022: A portion of a ride of an amusement park is to be modeled using two polynomials.The first section is the polynomial P(x) that covers the horizontal distance 100 feet, starts at the height of 60 feet and ends at 110 feet. The second section is polynomial Q(x) that covers the horizontal distance 50 feet, starts at the height of 110 feet and ends at 70 feet. Use the divided-difference method to find the (i) polynomials P(x), Q(x) and (ii) a polynomial R(x) for the complete portion of the ride. Construct the graph as well.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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
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