To find the highest point, we need to find the vertex of the quadratic equation h(t)=-16t^2 +224t+1152. The formula for finding the x-coordinate of the vertex is -b/2a. In this case, a=-16 and b=224. Let's calculate the x-coordinate of the vertex using the formula.

To find the vertex of the quadratic equation \( h(t) = -16t^2 + 224t + 1152 \), we use the formula for the x-coordinate of the vertex, which is given by:

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

In this equation, \( a = -16 \) and \( b = 224 \).

Let's plug in the values:

\[ t = -\frac{224}{2 \times -16} \]
\[ t = -\frac{224}{-32} \]
\[ t = 7 \]

Therefore, the x-coordinate (or in this context, the time \( t \)) of the vertex is \( t = 7 \).

Now, to find the highest point (the maximum height), we substitute \( t = 7 \) back into the original equation:

\[ h(7) = -16(7)^2 + 224(7) + 1152 \]
\[ h(7) = -16(49) + 224 \times 7 + 1152 \]
\[ h(7) = -784 + 1568 + 1152 \]
\[ h(7) = 1936 \]

Thus, the highest point is 1936. Therefore, the maximum height is 1936 units.

Would you like any more details or have any questions?

Here are 5 questions you can ask:
1. How do we determine the y-coordinate of the vertex of a quadratic function?
2. Can you explain why the formula for the vertex works?
3. What are some practical applications of finding the vertex in real-world scenarios?
4. How does the sign of the coefficient \(a\) affect the shape and vertex of the parabola?
5. Can you show the complete steps to solve another quadratic equation using the vertex formula?

**Tip:** When solving quadratic equations, always pay attention to the signs of the coefficients, as they significantly affect the shape and position of the graph.

Learn how to find the highest point (maximum height) of a quadratic equation h(t) = -16t^2 + 224t + 1152 using the vertex formula. Step-by-step solution with detailed calculations included.

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