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) Responses The initial position of the water bottle is (0,5.5) . The vertex of the throw is (1,16) . The maximum height the water bottle can reach is 16 feet. Sara will not be able to reach the bottle. The initial position of the water bottle is open paren 0 comma 5 point 5 close paren. The vertex of the throw is left parenthesis 1 comma 16 right parenthesis . The maximum height the water bottle can reach is 16 feet. Sara will not be able to reach the bottle. The initial position of the water bottle is (0,5.5) . The vertex of the throw is (1,21.5) . The maximum height the water bottle can reach is 21.5 feet. Sara will be able to reach the bottle. The initial position of the water bottle is open paren 0 comma 5 point 5 close paren. The vertex of the throw is open paren 1 comma 21 point 5 close paren. The maximum height the water bottle can reach is 21 point 5 feet. Sara will be able to reach the bottle. The initial position of the water bottle is (5.5,0) . The vertex of the throw is (21.5,1) . The maximum height the water bottle can reach is 21.5 feet. Sara will be able to reach the bottle. The initial position of the water bottle is open paren 5 point 5 comma 0 close paren. The vertex of the throw is left parenthesis 21.5 comma 1 right parenthesis . The maximum height the water bottle can reach is 21 point 5 feet. Sara will be able to reach the bottle. The vertex of the throw is (1,21) . The maximum height the water bottle can reach is 21 feet. Sara will be able to reach the bottle.

Answer 1

To solve the problem, let's analyze the scenario systematically.

1. Initial Position:

   • Lex tosses the water bottle from 5.5 feet above the ground. Therefore, the initial position of the water bottle is (0, 5.5).

2. Using the Function:

   • The height function of the water bottle is given by $h(t)=-16t^2+32t+5.5$, where $t$ is the time in seconds.

3. Finding the Maximum Height:

   • To find the maximum height, we need to determine the vertex of the parabola described by the height function. The formula for the vertex $t$ coordinate is $t=-\frac{b}{2a}$, where $a=-16$ and $b=32$.
   • Compute $t$: $$t=-\frac{32}{2\times -16}=\frac{32}{32}=1$$
   • Now substitute $t=1$ back into the original height function to find the maximum height: $$h(1)=-16(1{)}^2+32(1)+5.5=-16+32+5.5=21.5$$

4. Conclusion:

   • The maximum height the water bottle can reach is 21.5 feet.
   • Since Sara is 20 feet above the ground and the maximum height of the bottle (21.5 feet) is greater than 20 feet, Sara will be able to reach the bottle.

Thus, the correct response is: The initial position of the water bottle is (0, 5.5). The vertex of the throw is (1, 21.5). The maximum height the water bottle can reach is 21.5 feet. Sara will be able to reach the bottle.