To solve this problem, we want to find the maximum height of the ball and the time it takes to reach this height. The function that models the scenario is given as:
\[ h(t) = -16t^2 + 48t + 100 \]
This is a quadratic function of the form \( h(t) = at^2 + bt + c \), where:
- \( a = -16 \)
- \( b = 48 \)
- \( c = 100 \)
Key Feature of the Graph
The key feature that we are looking for in the problem is the vertex of the parabola represented by the function \( h(t) \). Since the parabola opens downward (because \( a < 0 \)), the vertex represents the maximum point on the graph, which corresponds to the maximum height of the ball.
Finding the Time to Reach Maximum Height
The time \( t \) at which the maximum height occurs can be found using the formula for the vertex of a quadratic function:
\[ t = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ t = -\frac{48}{2 \times -16} = -\frac{48}{-32} = \frac{48}{32} = \frac{3}{2} = 1.5 \text{ seconds} \]
Finding the Maximum Height
Now that we have the time at which the maximum height occurs, we can find the maximum height by substituting \( t = 1.5 \) back into the height function \( h(t) \):
\[ h(1.5) = -16(1.5)^2 + 48(1.5) + 100 \]
Calculating \( (1.5)^2 \):
\[ (1.5)^2 = 2.25 \]
Now substituting this value into the equation:
\[ h(1.5) = -16(2.25) + 48(1.5) + 100 \] \[ = -36 + 72 + 100 \] \[ = 36 + 100 = 136 \text{ feet} \]
Conclusion
- The maximum height the ball reaches is 136 feet.
- It takes 1.5 seconds to reach that height.
Thus, the key feature of the graph that the problem is asking you to find is the maximum height (the vertex of the parabola) and the time at which it occurs.
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