A ball thrown into the air from a building 100 ft high travels along a path described by y = -x^2/100 + 1x +100, where y is the height in feet and z is the horizontal distance from the building in feet. What is the maximum height the ball will reach?

To find the maximum height the ball will reach, we need to determine the vertex of the parabola represented by the equation

\[
y = -\frac{x^2}{100} + x + 100
\]

This is a quadratic equation in the form:

\[
y = ax^2 + bx + c
\]

where \( a = -\frac{1}{100} \), \( b = 1 \), and \( c = 100 \). In a quadratic function \( y = ax^2 + bx + c \), the x-coordinate of the vertex, which gives the maximum height in this case, is given by:

\[
x = -\frac{b}{2a}
\]

### Step 1: Find the x-coordinate of the vertex
Plugging in the values of \( a \) and \( b \):

\[
x = -\frac{1}{2 \cdot \left(-\frac{1}{100}\right)} = \frac{1}{\frac{2}{100}} = 50
\]

So, the maximum height occurs when \( x = 50 \) feet from the building.

### Step 2: Substitute \( x = 50 \) into the equation to find \( y \)
Now, substitute \( x = 50 \) back into the equation to find the corresponding \( y \)-value:

\[
y = -\frac{(50)^2}{100} + 1 \cdot 50 + 100
\]

Calculating each term:

1. \( -\frac{(50)^2}{100} = -\frac{2500}{100} = -25 \)
2. \( 1 \cdot 50 = 50 \)
3. \( +100 \)

Adding these together:

\[
y = -25 + 50 + 100 = 125
\]

### Conclusion
The maximum height the ball will reach is **125 feet**.

---

Would you like additional details or have any questions?

Here are 5 related questions to expand on this problem:
1. How do we interpret the path of the ball using the equation given?
2. What would change if the ball was thrown with a different initial velocity?
3. How would the maximum height be affected if the building were taller or shorter?
4. What is the significance of the vertex in a projectile motion problem?
5. How would we calculate the total distance traveled by the ball horizontally?

**Tip:** In projectile motion problems, the vertex represents the highest point the object will reach.

A ball thrown into the air from a building 100 ft high travels along a path described by y = -x^2/100 + 1x +100, where y is the height in feet and x is the horizontal distance from the building in feet. What is the maximum height the ball will reach?

Explore how to determine the maximum height of a ball thrown from a 100 ft high building, using the quadratic equation y = -x^2/100 + x + 100. This problem demonstrates key concepts in algebra and projectile motion suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation