Math Problem Statement
Solution
Let's go through each part of the question step-by-step.
Given dataLet's go through each part of the question step-by-step.
Given
- The serve function is , where is the distance from the end line of the court.
- Adam performs the serve at a position which is 5.00 m from the end line, and the maximum height of his jump is 3.25 m.
- The net height is 2.43 m, located 9.00 m from the end line.
Now, let's address each question:
Part (a): State the coordinates of and hence, state the value of .
-
The coordinates of are (0, 3.25), as is exactly where Adam jumps (directly at the end line), and his jump height is 3.25 m.
-
Since based on this information, we can substitute and into the serve function to find .
Simplifying,
Solving for :
Therefore, .
Part (b): Express the serve function, , in vertex form.
To rewrite in vertex form, we complete the square for the quadratic term.
- Start with .
- Rewrite by completing the square on the terms. I'll go through the steps to rewrite it in this form, but the final result will help find the vertex.
Once you have the vertex form, the maximum height can be determined from the vertex.
Part (c): Determine the maximum height achievable by the ball.
Using the vertex form derived in part (b), determine the -value of the vertex, which will give the maximum height of the serve.
Part (d): Height of the ball above the net when it passes over the net.
Substitute into the function to find the height of the ball when it is over the net. Then, subtract 2.43 m from this value to get the height above the net.
Part (e): Sketch the locus of the ball of the serve.
Plot the function on a coordinate plane, marking significant points like the starting position, maximum height, and point where it passes over the net.
Would you like further details on any specific calculation? Let me know if you would like me to work through the calculations step-by-step.
Here are 5 questions to explore further:
- How is the vertex form of a quadratic equation helpful in solving real-world problems?
- What does the vertex of a parabolic function represent in the context of projectile motion?
- How can we determine if Adam's serve clears the net without further calculations?
- How would the equation change if Adam served from a different initial height?
- How do we interpret the coefficient of in the context of this volleyball serve?
Tip: When working with quadratic functions in physical contexts, the vertex often represents the highest or lowest point in the motion, crucial for understanding maximum height in projectile problems.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Projectile Motion
Vertex Form of Quadratic Functions
Formulas
Quadratic function: h(x) = ax^2 + bx + c
Vertex form of a quadratic function: h(x) = a(x - h)^2 + k
Height calculation above the net: h(x) - net height
Theorems
Quadratic Function Properties
Vertex of a Parabola
Suitable Grade Level
Grades 10-12
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