A golf ball is struck by a​ 60-degree golf club at an initial velocity of 96 feet per second. The height of the golf ball in feet is given by the quadratic function h left parenthesis x right parenthesis equals negative StartFraction 16 x squared Over left parenthesis 48 right parenthesis squared EndFraction plus StartFraction 83.1 Over 48 EndFraction x
​, where x is the horizontal distance of the golf ball from the point of impact. What is the horizontal distance of the golf ball from the point of impact when the ball is at its maximum​ height? What is the maximum height obtained by the golf​ ball?
Question content area bottom
Part 1
The horizontal distance of the golf ball from the point of impact when the ball is at its maximum height is 
  
enter your response here feet.
​(Round to two decimal places as​ needed.) 

We are given the height function of the golf ball in terms of its horizontal distance \( x \), which is written as:

\[
h(x) = -\frac{16x^2}{48^2} + \frac{83.1}{48}x
\]

This quadratic function is in the form \( h(x) = ax^2 + bx + c \), where \( a = -\frac{16}{48^2} \) and \( b = \frac{83.1}{48} \). The goal is to find two things:

1. **The horizontal distance at maximum height.**
2. **The maximum height.**

### Step 1: Find the horizontal distance of the maximum height

For a quadratic function \( ax^2 + bx + c \), the horizontal distance at which the maximum (or minimum) occurs can be found using the formula for the vertex:

\[
x_{\text{max}} = \frac{-b}{2a}
\]

Here, \( a = -\frac{16}{48^2} \) and \( b = \frac{83.1}{48} \).

Let's calculate \( x_{\text{max}} \).

\[
x_{\text{max}} = \frac{-\frac{83.1}{48}}{2 \times \left(-\frac{16}{48^2}\right)} = \frac{-\frac{83.1}{48}}{-\frac{32}{48^2}} = \frac{83.1}{48} \times \frac{48^2}{32}
\]

Now simplify:

\[
x_{\text{max}} = \frac{83.1 \times 48}{32}
\]

\[
x_{\text{max}} = \frac{3988.8}{32} = 124.65 \, \text{feet}
\]

Thus, the horizontal distance of the golf ball from the point of impact when it reaches its maximum height is approximately **124.65 feet**.

### Step 2: Find the maximum height

To find the maximum height, substitute \( x_{\text{max}} = 124.65 \) back into the height function \( h(x) \).

\[
h(124.65) = -\frac{16(124.65)^2}{48^2} + \frac{83.1}{48} \times 124.65
\]

Now, let's calculate each term:

\[
124.65^2 = 15537.42
\]
\[
\frac{16 \times 15537.42}{48^2} = \frac{248598.72}{2304} = 107.92
\]
\[
\frac{83.1}{48} \times 124.65 = 2.73125 \times 124.65 = 340.43
\]

Now calculate the height:

\[
h(124.65) = -107.92 + 340.43 = 232.51 \, \text{feet}
\]

Thus, the maximum height obtained by the golf ball is approximately **232.51 feet**.

### Final Answers:
- The horizontal distance from the point of impact when the ball is at its maximum height is approximately **124.65 feet**.
- The maximum height obtained by the golf ball is approximately **232.51 feet**.

Would you like more details, or do you have any questions?

### 5 Related Questions:
1. How can we find the time of flight for the golf ball using a similar function?
2. What happens to the horizontal distance if the initial velocity increases?
3. How can we adjust the equation if the launch angle changes?
4. How do wind resistance and air drag affect the height and distance?
5. How does changing gravity (e.g., on the moon) alter the trajectory?

### Tip:
For any parabolic motion, the vertex formula \( x = \frac{-b}{2a} \) is essential for finding maximum height or distance.

A golf ball is struck by a 60-degree golf club at an initial velocity of 96 feet per second. The height of the golf ball in feet is given by the quadratic function h(x) = -(16x^2)/(48^2) + (83.1/48)x, where x is the horizontal distance of the golf ball from the point of impact. What is the horizontal distance of the golf ball from the point of impact when the ball is at its maximum height? What is the maximum height obtained by the golf ball?

This problem demonstrates how to find the horizontal distance and maximum height of a golf ball using a quadratic function h(x) = -(16x^2)/(48^2) + (83.1/48)x. By applying the vertex formula, students learn key algebraic concepts related to parabolic motion. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation