Question 1180642
Here's how to create a quadratic relation that models a basketball shot hitting the rim from 15 feet away:

**Understanding the Problem:**

* We're given the standard quadratic model for a basketball shot: h = -0.2d² + 3d + 6
    * h: height of the ball
    * d: distance from the shooter

* We need to create a new quadratic model where:
    * The shot is taken from 15 feet away (d = 15).
    * The ball hits the rim (we'll assume a standard rim height of 10 feet, so h = 10 when d = 15).

**Creating the New Quadratic Model:**

1. **General Form:** Start with the general form of a quadratic relation: h = ad² + bd + c

2. **Use the Given Information:**
    * We know the shot is from 15 feet and hits the 10-foot rim, so we have a point (d, h) = (15, 10).  Substitute this into the equation:
        10 = a(15)² + b(15) + c
        10 = 225a + 15b + c

    * To make the shot hit the rim at this distance, we need the vertex of the parabola to be at d = 15. The x-coordinate (in this case, d-coordinate) of the vertex of the parabola is given by -b/2a.  So:
        15 = -b / 2a
        -30a = b

3. **Choose a Value for 'a':**
    * We have some freedom here. Let's choose a value for 'a' that's different from the original model but still makes sense for a basketball shot.  A slightly smaller value for 'a' would make the shot arc higher. Let's try a = -0.15.

4. **Solve for 'b' and 'c':**
    * Using -30a = b and a = -0.15, we get:
        b = -30 * (-0.15) = 4.5

    * Substitute a = -0.15 and b = 4.5 into the equation 10 = 225a + 15b + c:
        10 = 225(-0.15) + 15(4.5) + c
        10 = -33.75 + 67.5 + c
        c = -23.75

**The New Quadratic Model:**

* h = -0.15d² + 4.5d - 23.75

This model represents a basketball shot taken from 15 feet away that hits the rim of the basketball net.

**Explanation:**

* We used the given information (distance and height of the rim) to create an equation with the general form of a quadratic relation.
* We used the fact that the vertex of the parabola should be at the rim (d = 15) to relate 'a' and 'b'.
* By choosing a value for 'a', we could solve for 'b' and 'c' to complete the model.