A golfer hits a golf ball toward the hole 55 yards away. The ball follows a parabolic path given by the function h(x)=−1/25x(x−45) , where h(x) is the height of the golf ball and x is the horizontal distance it has traveled. Solve the equation to determine how far the ball will have traveled when it hits the ground. How far will the ball need to roll to make it to the hole?

Question

A golfer hits a golf ball toward the hole 55 yards away. The ball follows a parabolic path given by the function  h(x)=−1/25x(x−45) , where  h(x)  is the height of the golf ball and x is the horizontal distance it has traveled. Solve the equation to determine how far the ball will have traveled when it hits the ground. How far will the ball need to roll to make it to the hole?

GPT-4o mini · Answer

To determine how far the golf ball will travel before hitting the ground, we need to find the values of \(x\) for which the height \(h(x)\) equals zero. The function given is

\[ h(x) = -\frac{1}{25}x(x - 45). \]

Setting \(h(x) = 0\) gives us the equation:

\[ -\frac{1}{25} x(x - 45) = 0. \]

We can multiply both sides by -25 to simplify:

\[ x(x - 45) = 0. \]

This equation is satisfied when either \(x = 0\) or \(x - 45 = 0\). Solving these gives us:

\[ x = 0 \quad \text{or} \quad x = 45. \]

The value \(x = 0\) corresponds to the initial point where the ball is struck, and \(x = 45\) is where it hits the ground. Thus, the ball will hit the ground after traveling 45 yards.

Next, we want to determine how far the ball will need to roll to reach the hole, which is 55 yards away. The distance the ball needs to roll after hitting the ground is:

\[ 55 \text{ yards} - 45 \text{ yards} = 10 \text{ yards}. \]

Thus, the ball will need to roll 10 yards to make it to the hole.

In summary:

The ball travels 45 yards before hitting the ground.
The ball needs to roll 10 yards to reach the hole.