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The problem given is about the motion of Romeo, who is thrown from a point above the ground. The height \( h \) at ground level is given by the equation:

\[ h = -8t^2 + 48t \]

where \( h \) is the height in meters and \( t \) is the time in seconds.

### 1. Finding the Maximum Height:
To find the maximum height, we need to find the vertex of the quadratic equation \( h = -8t^2 + 48t \). The time \( t \) at which the maximum height occurs can be found using the formula:

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

For the given equation \( h = -8t^2 + 48t \), the coefficients are:
- \( a = -8 \)
- \( b = 48 \)

Substitute these values into the formula:

\[ t = -\frac{48}{2(-8)} = \frac{48}{16} = 3 \text{ seconds} \]

So, the time to reach maximum height is 3 seconds.

Next, we calculate the maximum height by substituting \( t = 3 \) into the equation for \( h \):

\[ h = -8(3)^2 + 48(3) \]
\[ h = -8(9) + 144 \]
\[ h = -72 + 144 = 72 \text{ meters} \]

### 2. Time to Reach Maximum Height:
The time to reach the maximum height is already found to be **3 seconds**.

### Summary
- **Maximum Height**: 72 meters
- **Time to Reach Maximum Height**: 3 seconds

Would you like more details or have any questions?

#### Related Questions:
1. What is the total time for Romeo to return to the ground?
2. What is the initial velocity of Romeo's throw?
3. What is the value of acceleration due to gravity in this scenario?
4. How high is Romeo above the ground at \( t = 0 \)?
5. Can you derive the equation of motion from the given data?

**Tip:** For any quadratic equation of the form \( ax^2 + bx + c \), the vertex formula can quickly give you the maximum or minimum value, depending on the direction of the parabola.

Romeo is thrown from a point above the ground. The height at ground level is given by the equation h = -8t^2 + 48t, where h is the height in meters and t is the time in seconds. What is the maximum height he can reach? How long will it take to reach the maximum height?

Learn how to calculate the maximum height of a projectile thrown into the air using the quadratic equation h = -8t^2 + 48t. Find out how long it takes to reach the maximum height with step-by-step calculations. Ideal for students in Grades 9-10.

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