To solve this problem, let's first clarify the initial position of the water bottle and the maximum height it can reach.
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Initial Position of the Water Bottle: Lex tosses the bottle from a height of 5.5 feet above the ground. Since he is 5.5 feet off the ground, the initial position \( h_0 \) of the water bottle when it is tossed is: \[ h_0 = 5.5 , \text{feet} \]
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Finding the Maximum Height: The height function given is: \[ h(t) = -16t^2 + 32t \] This is a quadratic equation where \( h(t) \) represents the height of the bottle at time \( t \).
To find the maximum height, we need to determine the vertex of this parabola. For a quadratic equation in the form \( at^2 + bt + c \), the time \( t \) at which the maximum height occurs can be calculated using the formula: \[ t = -\frac{b}{2a} \] In this function \( a = -16 \) and \( b = 32 \): \[ t = -\frac{32}{2 \times -16} = -\frac{32}{-32} = 1 , \text{second} \]
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Calculating the Maximum Height: Now, we will substitute \( t = 1 \) back into the height function to find the maximum height: \[ h(1) = -16(1)^2 + 32(1) = -16 + 32 = 16 , \text{feet} \]
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Final Comparison: Sara is positioned at a height of 20 feet, and the maximum height that Lex can toss the water bottle is 16 feet. Thus, the maximum height of the tossed water bottle is 16 feet, which is below the height of 20 feet where Sara is located.
Conclusion: Sara will not be able to reach the bottle since the maximum height of the bottle (16 feet) is lower than Sara's height (20 feet).
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