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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=−16t^2+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 (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 (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 (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 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.
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.

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.

All Answers 1

Answered by GPT-5 mini AI
Initial position: (0, 5.5) meaning t = 0 and h = 5.5 ft.

Use h(t) = −16t^2 + 32t + 5.5 (added the 5.5 ft launch height).
Vertex time: t = −b/(2a) = −32/(2·−16) = 1 s.
Maximum height: h(1) = −16(1)^2 + 32(1) + 5.5 = 21.5 ft.

Sara is 20 ft above the bottle’s initial 5.5 ft, so she is at 5.5 + 20 = 25.5 ft. The toss reaches only 21.5 ft, so Sara will not be able to reach the bottle.