Mathematico | Mathematico

A golfer chips a golf ball from ground level. The path of the golf ball follows the shape of a parabola.

Parabola

As it descends, it just clears the top of a fence. The fence is $2 m$ tall and is at a distance of $8 m$ from the golfer. The ball then travels a further $3 m$ before hitting the ground at the same horizontal level as it was when first struck.

You are given that the path of the ball may be modeled by

h = k x (x - d)

where $h$ is the height of the ball, $x$ is the horizontal distance traveled, and $d, k$ are non-zero constants.

Find the values of $d$ and $k$ .
Given that the ball just clears a shrub of height $1.5 m$ as it is ascending, find the horizontal distance from the golfer to the shrub.
Find the greatest height reached by the golf ball.

Let the answers to (b) and (c) be $x$ and $y$ . Give the value of

\sqrt{24 x y}

Hint

Overall, the diagram for this problem should look like this:

Annotated parabola

Hint

To find $d$ , notice that $x = 11$ is one of the roots of the parabola.

To find $k$ , notice that the point $(8, 2)$ lies on the parabola. Substitute this into the equation of the parabola and rearrange.

Solution

The diagram showing all the given information looks like this:

Annotated parabola

When $x = 11, h = 0$ , and so
$\begin{aligned} 0 & = 11 k (11 - d) \\ 11 - d & = 0 \\ d & = 11 \end{aligned}$
We also know that when $x = 8, h = 2$ so
$\begin{aligned} h & = k x (x - 11) \\ 2 & = 8 k (8 - 11) \\ 2 & = - 24 k \\ k & = - \frac{1}{12} \end{aligned}$
Letting $h = 1.5$ , we find
$\begin{aligned} 1.5 & = - \frac{1}{12} x (x - 11) \\ - 18 & = x (x - 11) \\ = x^{2} - 11 x \\ x^{2} - 11 x + 18 & = 0 \\ (x - 9) (x - 2) & = 0 \\ x & \in {2, 9} \end{aligned}$
However, since the ball is ascending we must have $x = 2$ .
Consider that the roots are $x = 0$ and $x = 11$ , by symmetry the highest point will occur when $x = 5.5$ , so
$\begin{aligned} h & = - \frac{1}{12} \times 5.5 \times (5.5 - 11) \\ = \frac{121}{48} \end{aligned}$