- Each bounce is modelled by the equation $$ y=-\frac{1}{2}(x-p)^{2}+q $$. - If the ball continued to bounce, it would eventually converge with the horizontal 16 metres from point $$ A $$. (1) Determine the length of AB. IEB Copyright 20222 NATIONAL SENIOR CERTIFICATE: MATHEMATICs: PAPERI (2) What is the maximum height reached by the ball between points B and C?

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Given:
- The equation of each bounce is $$ y = -\frac{1}{2}(x-p)^{2} + q $$.
- The ball will eventually converge with the horizontal 16 meters from point $$ A $$.

To solve this problem, we need to find the length of AB and the maximum height reached by the ball between points B and C.

1. Determine the length of AB:
Since the ball will eventually converge with the horizontal 16 meters from point $$ A $$, we can assume that the point of convergence is the vertex of the parabola. The vertex form of a parabola is given by $$ y = a(x-h)^{2} + k $$, where (h, k) is the vertex of the parabola.

Comparing the given equation $$ y = -\frac{1}{2}(x-p)^{2} + q $$ with the vertex form, we can see that the vertex is at (p, q). Since the ball will converge with the horizontal 16 meters from point $$ A $$, the x-coordinate of the vertex is 16 meters from point $$ A $$. Therefore, the length of AB is 16 meters.

2. Find the maximum height reached by the ball between points B and C:
To find the maximum height reached by the ball between points B and C, we need to find the vertex of the parabola. The vertex of the parabola is given by (p, q), where p is the x-coordinate of the vertex and q is the y-coordinate of the vertex.

Since the length of AB is 16 meters, the x-coordinate of the vertex is 16 meters from point $$ A $$. Therefore, the x-coordinate of the vertex is 16 meters.

Substitute the x-coordinate of the vertex into the equation $$ y = -\frac{1}{2}(x-p)^{2} + q $$ to find the y-coordinate of the vertex, which represents the maximum height reached by the ball between points B and C.
The length of AB is 16 meters. The maximum height reached by the ball between points B and C is the y-coordinate of the vertex, which can be found by substituting the x-coordinate of the vertex into the equation $$ y = -\frac{1}{2}(x-p)^{2} + q $$.

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