A golf ball is struck by a​ 60-degree golf club at an initial velocity of 

8686 

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 43 right parenthesis squared EndFraction plus StartFraction 74.5 Over 43 EndFraction xh(x)=−16x2(43)2+74.543x​, 

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.) 

To find the horizontal distance \( x \) of the golf ball from the point of impact when the ball is at its maximum height and the maximum height itself, we start by analyzing the given quadratic function:

\[ h(x) = -\frac{16x^2}{43^2} + \frac{74.5}{43}x \]

This quadratic function is in the form \( h(x) = ax^2 + bx + c \) where:
\[ a = -\frac{16}{43^2} \]
\[ b = \frac{74.5}{43} \]
\[ c = 0 \] (since there is no constant term)

### Part 1: Finding the Horizontal Distance at Maximum Height

For a quadratic function \( h(x) = ax^2 + bx + c \), the vertex (maximum or minimum point) occurs at:
\[ x = -\frac{b}{2a} \]

Substitute \( a \) and \( b \) into the formula:
\[ a = -\frac{16}{43^2} = -\frac{16}{1849} \]
\[ b = \frac{74.5}{43} \]

\[ x = -\frac{\frac{74.5}{43}}{2 \left(-\frac{16}{1849}\right)} \]

Simplify the expression:
\[ x = \frac{\frac{74.5}{43}}{\frac{32}{1849}} \]
\[ x = \frac{74.5 \cdot 1849}{43 \cdot 32} \]
\[ x = \frac{137820.5}{1376} \]
\[ x \approx 100.13 \]

So, the horizontal distance \( x \) from the point of impact when the golf ball is at its maximum height is approximately \( 100.13 \) feet.

### Part 2: Finding the Maximum Height

Substitute \( x = 100.13 \) back into the height function \( h(x) \):
\[ h(100.13) = -\frac{16 (100.13)^2}{43^2} + \frac{74.5 \cdot 100.13}{43} \]

Calculate each term:
\[ \frac{16 (100.13)^2}{43^2} = \frac{16 \cdot 10026.0169}{1849} \approx 86.782 \]
\[ \frac{74.5 \cdot 100.13}{43} \approx 173.581 \]

Now, combine the terms:
\[ h(100.13) = 173.581 - 86.782 \]
\[ h(100.13) \approx 86.80 \]

So, the maximum height of the golf ball is approximately \( 86.80 \) feet.

### Summary

- The horizontal distance of the golf ball from the point of impact when the ball is at its maximum height is approximately \( 100.13 \) feet.
- The maximum height obtained by the golf ball is approximately \( 86.80 \) feet.

Would you like more details or have any questions about this solution?

Here are some related questions you might find interesting:
1. How would the problem change if the angle of the golf club was different?
2. What is the total time the golf ball spends in the air before hitting the ground?
3. How do you find the horizontal range of the golf ball?
4. How does the initial velocity affect the maximum height and horizontal distance?
5. What is the significance of the coefficient -16 in the height function?
6. How would wind resistance alter the golf ball's trajectory?
7. How can we determine the landing point of the golf ball?
8. What assumptions are made in this projectile motion model?

**Tip:** In projectile motion problems, converting units when necessary and carefully handling the trigonometric components of the initial conditions (like the angle of launch) can help ensure accurate calculations.

Learn how to find the horizontal distance and maximum height of a golf ball struck at an angle using a quadratic function. Detailed step-by-step solution provided with calculations for both horizontal distance and maximum height.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation