Determine the maximum height for an object launched with the function h(t)=-16t^2+360t

To determine the maximum height of an object launched with the function \( h(t) = -16t^2 + 360t \), we need to find the vertex of this parabolic equation, since the vertex will give us the maximum height (as the coefficient of \( t^2 \) is negative, indicating a parabola that opens downwards).

The vertex form of a parabola \( ax^2 + bx + c \) is given by the vertex \((h, k)\), where \( h = -\frac{b}{2a} \). In our case, the function is:

\[ h(t) = -16t^2 + 360t \]

Here, \( a = -16 \) and \( b = 360 \).

First, find the time \( t \) at which the maximum height occurs:

\[ t = -\frac{b}{2a} = -\frac{360}{2 \cdot -16} = \frac{360}{32} = 11.25 \]

Now, substitute \( t = 11.25 \) back into the function \( h(t) \) to find the maximum height:

\[ h(11.25) = -16(11.25)^2 + 360(11.25) \]

Calculate \( (11.25)^2 \):

\[ (11.25)^2 = 126.5625 \]

Now substitute and calculate:

\[ h(11.25) = -16 \cdot 126.5625 + 360 \cdot 11.25 \]
\[ h(11.25) = -2025 + 4050 \]
\[ h(11.25) = 2025 \]

Therefore, the maximum height of the object is \( 2025 \) feet.

Would you like more details or have any questions?

Here are 5 questions you might consider asking:
1. How do you determine the vertex of a parabola?
2. Can you explain why the vertex represents the maximum height in this context?
3. What are the units used in this problem?
4. How does the coefficient of \( t^2 \) affect the direction of the parabola?
5. Can this method be used for any quadratic function?

**Tip:** Always check the direction of the parabola (whether it opens upwards or downwards) to determine if you're finding a maximum or minimum value.

Learn how to determine the maximum height of an object launched with the quadratic function h(t) = -16t^2 + 360t. Understand the vertex form of a parabola and how it applies to finding maximum heights. Suitable for high school students studying quadratic equations and parabolas.

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