The Center Of An Ice Rink Is Located At \[$(0,0)\$\] On A Coordinate System Measured In Feet. Sandi Skates Along A Path That Can Be Modeled By The Equation \[$y=0.08x^2-1.6x+13\$\].David Starts At \[$(20,13)\$\] And Skates Along
The Center of an Ice Rink: A Mathematical Exploration of Sandi's Skating Path
In the world of mathematics, coordinate geometry plays a vital role in understanding various real-world phenomena. In this article, we will delve into the fascinating realm of coordinate geometry and explore the path of Sandi's skating on an ice rink. The center of the ice rink is located at (0,0) on a coordinate system measured in feet. Sandi's path can be modeled by the equation y = 0.08x^2 - 1.6x + 13. We will analyze this equation and determine the nature of Sandi's skating path.
The equation y = 0.08x^2 - 1.6x + 13 represents a quadratic function, which is a polynomial of degree two. The general form of a quadratic function is ax^2 + bx + c, where a, b, and c are constants. In this case, a = 0.08, b = -1.6, and c = 13.
To understand the nature of Sandi's skating path, we need to analyze the graph of the equation. The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola opens upward if a > 0 and downward if a < 0.
To graph the equation, we can use the following steps:
- Determine the x-intercepts of the parabola by setting y = 0 and solving for x.
- Determine the y-intercept of the parabola by setting x = 0 and solving for y.
- Use the x-intercepts and y-intercept to draw the parabola.
Let's perform these steps to graph the equation.
Finding the X-Intercepts
To find the x-intercepts, we set y = 0 and solve for x:
0 = 0.08x^2 - 1.6x + 13
We can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 0.08, b = -1.6, and c = 13. Plugging these values into the quadratic formula, we get:
x = (1.6 ± √((-1.6)^2 - 4(0.08)(13))) / (2(0.08)) x = (1.6 ± √(2.56 - 4.32)) / 0.16 x = (1.6 ± √(-1.76)) / 0.16
Since the square root of a negative number is not a real number, we conclude that the equation has no real x-intercepts.
Finding the Y-Intercept
To find the y-intercept, we set x = 0 and solve for y:
y = 0.08(0)^2 - 1.6(0) + 13 y = 13
Therefore, the y-intercept is (0, 13).
Graphing the Parabola
Since the equation has no real x-intercepts, the parabola does not intersect the x-axis. The parabola opens upward since a > 0. The y-intercept is (0, 13), which is the highest point on the parabola.
In conclusion, Sandi's skating path can be modeled by the equation y = 0.08x^2 - 1.6x + 13. The graph of this equation is a parabola that opens upward and has no real x-intercepts. The y-intercept is (0, 13), which is the highest point on the parabola.
David starts at (20, 13) and skates along the parabola. To find David's path, we need to find the equation of the line that passes through the point (20, 13) and is tangent to the parabola.
Finding the Equation of the Tangent Line
To find the equation of the tangent line, we need to find the slope of the line. The slope of the tangent line is equal to the derivative of the parabola at the point of tangency.
Let's find the derivative of the parabola:
y' = d/dx (0.08x^2 - 1.6x + 13) y' = 0.16x - 1.6
Now, we need to find the point of tangency. Since the parabola has no real x-intercepts, the point of tangency must be at x = 20.
y' = 0.16(20) - 1.6 y' = 3.2 - 1.6 y' = 1.6
Therefore, the slope of the tangent line is 1.6.
Finding the Equation of the Tangent Line
Now that we have the slope of the tangent line, we can find its equation. The equation of a line with slope m and passing through the point (x1, y1) is given by:
y - y1 = m(x - x1)
In this case, m = 1.6, x1 = 20, and y1 = 13. Plugging these values into the equation, we get:
y - 13 = 1.6(x - 20) y - 13 = 1.6x - 32 y = 1.6x - 19
Therefore, the equation of the tangent line is y = 1.6x - 19.
In conclusion, David's skating path can be modeled by the equation y = 1.6x - 19. This equation represents a line with slope 1.6 and passing through the point (20, 13).
Let's compare the paths of Sandi and David.
Sandi's path is modeled by the equation y = 0.08x^2 - 1.6x + 13, which is a parabola that opens upward and has no real x-intercepts. The y-intercept is (0, 13), which is the highest point on the parabola.
David's path is modeled by the equation y = 1.6x - 19, which is a line with slope 1.6 and passing through the point (20, 13).
Comparison of the Paths
Let's compare the two paths.
Sandi's path is a parabola that opens upward, while David's path is a line with slope 1.6. The parabola has no real x-intercepts, while the line passes through the point (20, 13).
In conclusion, Sandi's and David's paths are different. Sandi's path is a parabola that opens upward, while David's path is a line with slope 1.6. The parabola has no real x-intercepts, while the line passes through the point (20, 13).
In this article, we explored the paths of Sandi and David on an ice rink. Sandi's path is modeled by the equation y = 0.08x^2 - 1.6x + 13, which is a parabola that opens upward and has no real x-intercepts. David's path is modeled by the equation y = 1.6x - 19, which is a line with slope 1.6 and passing through the point (20, 13). We compared the two paths and found that they are different.
Q&A: Understanding Sandi's and David's Skating Paths
In our previous article, we explored the paths of Sandi and David on an ice rink. Sandi's path is modeled by the equation y = 0.08x^2 - 1.6x + 13, which is a parabola that opens upward and has no real x-intercepts. David's path is modeled by the equation y = 1.6x - 19, which is a line with slope 1.6 and passing through the point (20, 13). In this article, we will answer some frequently asked questions about Sandi's and David's skating paths.
Q: What is the equation of Sandi's skating path?
A: The equation of Sandi's skating path is y = 0.08x^2 - 1.6x + 13.
Q: What is the shape of Sandi's skating path?
A: Sandi's skating path is a parabola that opens upward.
Q: Does Sandi's skating path intersect the x-axis?
A: No, Sandi's skating path does not intersect the x-axis.
Q: What is the y-intercept of Sandi's skating path?
A: The y-intercept of Sandi's skating path is (0, 13).
Q: What is the equation of David's skating path?
A: The equation of David's skating path is y = 1.6x - 19.
Q: What is the shape of David's skating path?
A: David's skating path is a line with slope 1.6.
Q: Does David's skating path intersect the x-axis?
A: Yes, David's skating path intersects the x-axis at x = 20.
Q: What is the y-intercept of David's skating path?
A: The y-intercept of David's skating path is (0, -19).
Q: How do Sandi's and David's skating paths compare?
A: Sandi's skating path is a parabola that opens upward, while David's skating path is a line with slope 1.6. The parabola has no real x-intercepts, while the line passes through the point (20, 13).
Q: What is the significance of the y-intercept of Sandi's skating path?
A: The y-intercept of Sandi's skating path is (0, 13), which is the highest point on the parabola.
Q: What is the significance of the slope of David's skating path?
A: The slope of David's skating path is 1.6, which represents the rate of change of the line.
Q: Can you provide a visual representation of Sandi's and David's skating paths?
A: Yes, we can provide a visual representation of Sandi's and David's skating paths using a graph.
In this article, we answered some frequently asked questions about Sandi's and David's skating paths. We hope that this article has provided a better understanding of the paths of Sandi and David on an ice rink.
Here is a visual representation of Sandi's and David's skating paths:
Graph of Sandi's Skating Path
The graph of Sandi's skating path is a parabola that opens upward. The y-intercept is (0, 13), which is the highest point on the parabola.
Graph of David's Skating Path
The graph of David's skating path is a line with slope 1.6. The line passes through the point (20, 13).
Here is a comparison of Sandi's and David's skating paths:
| Sandi's Skating Path | David's Skating Path | |
|---|---|---|
| Equation | y = 0.08x^2 - 1.6x + 13 | y = 1.6x - 19 |
| Shape | Parabola that opens upward | Line with slope 1.6 |
| X-Intercepts | No real x-intercepts | x = 20 |
| Y-Intercept | (0, 13) | (0, -19) |
We hope that this article has provided a better understanding of the paths of Sandi and David on an ice rink.