To solve this problem, we need to determine if Lex can toss the water bottle to reach Sara at her height. Let’s break everything down step by step.
Step 1: Understand the initial position of the water bottle.
Lex tosses the bottle from a height of 5.5 feet above the ground, and Sara is at a height of: \[ \text{Sara's height} = 5.5 \text{ feet} + 20 \text{ feet} = 25.5 \text{ feet} \]
Step 2: Analyze the function provided.
The function given is: \[ h(t) = -16t^2 + 32t \]
This is a standard form of the quadratic equation representing the height of the bottle over time \( t \), where:
- The coefficient of \( t^2 \) represents the effect of gravity (specifically in feet per second squared),
- The first term \( 32t \) is the initial velocity of the bottle (in feet per second).
Step 3: Determine the maximum height of the water bottle.
To find the maximum height of the bottle, we first need to find the time at which it reaches maximum height. The maximum height for a quadratic function \( h(t) = at^2 + bt + c \) occurs at: \[ t = -\frac{b}{2a} \]
For our function:
- \( a = -16 \)
- \( b = 32 \)
Now, substituting these values gives: \[ t = -\frac{32}{2 \cdot -16} = -\frac{32}{-32} = 1 \text{ second} \]
Step 4: Calculate the maximum height.
Now we will use this time value to find the maximum height: \[ h(1) = -16(1)^2 + 32(1) = -16 + 32 = 16 \text{ feet} \]
Step 5: Determine if the bottle reaches Sara.
The water bottle reaches a maximum height of 16 feet. Lex tosses the bottle from a height of 5.5 feet, which means the highest point it reaches during the toss is: \[ \text{Maximum height} = 5.5 \text{ feet} + 16 \text{ feet} = 21.5 \text{ feet} \]
Step 6: Conclusion.
Since Sara is at 25.5 feet and the maximum height of the water bottle is 21.5 feet, the water bottle will not reach Sara. Thus, Lex will not be able to toss the bottle up to Sara.