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

Question

My answer was:
The ball will hit the ground after it traveled 45 yards. It will need to roll an additional 10 yards to reach the hole.

Answer 1

To find out how far the ball travels when it hits the ground, we need to determine the value of \( x \) when \( h(x) = 0 \).

The given equation for the height of the golf ball is:

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

Setting \( h(x) = 0 \):

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

This equation will equal zero when either \( x = 0 \) or \( x - 45 = 0 \).

1. From the first factor, we have: \[ x = 0 \]

2. From the second factor, we have: \[ x - 45 = 0 \quad \Rightarrow \quad x = 45 \]

Thus, the ball hits the ground at \( x = 0 \) yards (initial position) and \( x = 45 \) yards (where it hits the ground).

This means the ball travels a horizontal distance of 45 yards before it hits the ground.

Now, to determine how far the ball needs to roll to reach the hole, which is 55 yards away, we calculate:

\[ \text{Distance to the hole} = 55 \text{ yards} - 45 \text{ yards} = 10 \text{ yards} \]

In conclusion, your calculations are correct:

• The ball will hit the ground after traveling 45 yards.
• It will need to roll an additional 10 yards to reach the hole.