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**"The approximate height of a rock in meters, thrown upwards off a cliff, is \( h(t) = -5t^2 + 10t + 180 \) where \( t \) is time in seconds. What is the maximum height of the rock?"**

This problem requires finding the maximum height of the rock, which involves determining the vertex of the quadratic function \( h(t) = -5t^2 + 10t + 180 \).

### Steps to Solve:

The height function \( h(t) \) is a quadratic function of the form \( h(t) = at^2 + bt + c \), where:
- \( a = -5 \)
- \( b = 10 \)
- \( c = 180 \)

For a quadratic function \( h(t) = at^2 + bt + c \), the time \( t \) at which the maximum height occurs can be found using the formula:

\[
t_{\text{max}} = -\frac{b}{2a}
\]

Substitute the values of \( a \) and \( b \):

\[
t_{\text{max}} = -\frac{10}{2(-5)} = \frac{10}{10} = 1 \text{ second}
\]

Now, substitute \( t_{\text{max}} = 1 \) second back into the height function to find the maximum height:

\[
h(1) = -5(1)^2 + 10(1) + 180
\]

\[
h(1) = -5(1) + 10 + 180
\]

\[
h(1) = -5 + 10 + 180 = 185 \text{ meters}
\]

### Answer:
The maximum height of the rock is **185 meters**.

Would you like more details or have any further questions?

### Related Questions:
1. What is the significance of the coefficient \( a \) being negative in the quadratic function?
2. How can you determine the time when the rock hits the ground?
3. What would happen to the maximum height if the initial velocity of the rock was doubled?
4. How does the maximum height change if the rock is thrown from a lower cliff?
5. Can you derive the general formula for the vertex of any quadratic function?

### Tip:
When dealing with quadratic functions, remember that the vertex represents the maximum or minimum point. The sign of the coefficient \( a \) determines whether the parabola opens upwards (minimum) or downwards (maximum).

Discover how to find the maximum height of a rock thrown upwards from a cliff using a quadratic function. Solve the problem step-by-step with detailed calculations and formulas. Ideal for high school students learning quadratic functions and vertex calculations.

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