Question

Sara and Lex are hiking in the Rocky Mountain National Forest when Sara finds a ledge to climb. Once she gets to the top, she realizes she left her water bottle 20 feet below. She asks Lex to toss it up to her. Lex tosses the bottle from 5.5 feet above the ground. What is the initial position of the water bottle, and what is the maximum height she can toss the water bottle so it will reach Sara? Use the function h=−16t2+32t
to create a function to determine the point at which the bottle reaches its maximum height. Will Sara be able to reach the bottle?(1 point)

Answer 1

To solve this problem, let's first clarify the initial position of the water bottle and the maximum height it can reach.

1. **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:
\[
h_0 = 5.5 \, \text{feet}
\]

2. **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 \( b = 32 \):
\[
t = -\frac{32}{2 \times -16} = -\frac{32}{-32} = 1 \, \text{second}
\]

3. **Calculating the Maximum Height:**
Now, we will substitute \( t = 1 \) back into the height function to find the maximum height:
\[
h(1) = -16(1)^2 + 32(1) = -16 + 32 = 16 \, \text{feet}
\]

4. **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).