Given:
The height of a basketball is given by the function:
[tex]h(x)=-0.5x^2+3x+6[/tex]
where x is the horizontal distance from where it is thrown.
To find:
How far away from the basket should the player stand in order for the ball to go in the basket (10 feet high) on its way down.
Solution:
We have,
[tex]h(x)=-0.5x^2+3x+6[/tex]
Putting [tex]h(x)=10[/tex], we get
[tex]10=-0.5x^2+3x+6[/tex]
[tex]10+\dfrac{1}{2}x^2-3x-6=0[/tex]
[tex]\dfrac{1}{2}x^2-3x+4=0[/tex]
Multiply both sides by 2.
[tex]x^2-6x+8=0[/tex]
Splitting the middle term, we get
[tex]x^2-4x-2x+8=0[/tex]
[tex]x(x-4)-2(x-4)=0[/tex]
[tex](x-2)(x-4)=0[/tex]
[tex]x=2,4[/tex]
In the given function the leading coefficient is negative, so the given function represents a downward parabola. It means, first the function is increasing after that the function is decreasing.
So, the value of the function is 10 at [tex]x=2[/tex] (its way up) and at [tex]x=4[/tex] (its way down.
Therefore, the player should stand 4 units away from the basket in order for the ball to go in the basket (10 feet high) on its way down.
