Question
A ball is thrown upward from the top of a 30-foot-high building with an initial velocity of 60 ft/sec
. The maximum height of the ball, h, at any time, t, is modeled by the equation h≤−16t2+60t+30
.
Which of the following graphs represents this equation?
(1 point)
Responses
The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 60 to 90 by fifteens. The graph of a solid curve opens upward and goes through the points left parenthesis negative 4 comma 46 right parenthesis; left parenthesis negative 2 comma negative 26 right parenthesis; and left parenthesis 0 comma 30 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 60 to 90 by fifteens. The graph of a solid curve opens upward and goes through the points left parenthesis negative 4 comma 46 right parenthesis; left parenthesis negative 2 comma negative 26 right parenthesis; and left parenthesis 0 comma 30 right parenthesis. There is shading below the curve.
The coordinate grid has t axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a dashed curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has t axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a dashed curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading below the curve.
The coordinate grid has t axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has t axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading below the curve.
The coordinate grid has t-axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading above the curve.
Image with alt text: The coordinate grid has t-axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading above the curve.
. The maximum height of the ball, h, at any time, t, is modeled by the equation h≤−16t2+60t+30
.
Which of the following graphs represents this equation?
(1 point)
Responses
The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 60 to 90 by fifteens. The graph of a solid curve opens upward and goes through the points left parenthesis negative 4 comma 46 right parenthesis; left parenthesis negative 2 comma negative 26 right parenthesis; and left parenthesis 0 comma 30 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 60 to 90 by fifteens. The graph of a solid curve opens upward and goes through the points left parenthesis negative 4 comma 46 right parenthesis; left parenthesis negative 2 comma negative 26 right parenthesis; and left parenthesis 0 comma 30 right parenthesis. There is shading below the curve.
The coordinate grid has t axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a dashed curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has t axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a dashed curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading below the curve.
The coordinate grid has t axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has t axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading below the curve.
The coordinate grid has t-axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading above the curve.
Image with alt text: The coordinate grid has t-axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading above the curve.
Answers
GPT-4o mini
The equation provided, \( h = -16t^2 + 60t + 30 \), is a quadratic equation that opens downward because the coefficient of \( t^2 \) is negative (-16). This means the maximum height of the ball will be at the vertex of the parabola.
Let's analyze the features of the given options:
1. **Direction of the Curve**: The graph should open downward.
2. **Initial Height**: When \( t = 0 \), \( h = 30 \). Therefore, the graph should pass through the point (0, 30).
3. **Graph Characteristics**: Since \( h \) represents height, it cannot be negative; thus, we should consider the relevant points where \( h \) is at least 0.
Based on these characteristics, the correct graph should show a downward-opening curve that passes through (0, 30) and does not have any positive height values significantly below that.
Among the options provided, the only graph that fulfills these criteria is:
- **The one that includes a solid curve that opens downward and goes through the points (0, 30)**.
Specifically, the correct answer would be:
> **The coordinate grid has t axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading below the curve.**
This description matches the features of the quadratic and its characteristics, specifically emphasizing that it correctly represents the physical situation described.
Let's analyze the features of the given options:
1. **Direction of the Curve**: The graph should open downward.
2. **Initial Height**: When \( t = 0 \), \( h = 30 \). Therefore, the graph should pass through the point (0, 30).
3. **Graph Characteristics**: Since \( h \) represents height, it cannot be negative; thus, we should consider the relevant points where \( h \) is at least 0.
Based on these characteristics, the correct graph should show a downward-opening curve that passes through (0, 30) and does not have any positive height values significantly below that.
Among the options provided, the only graph that fulfills these criteria is:
- **The one that includes a solid curve that opens downward and goes through the points (0, 30)**.
Specifically, the correct answer would be:
> **The coordinate grid has t axis labeled from negative 5 to 5 and h axis labeled from negative 60 to 90 in intervals of 15. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 46 right parenthesis; left parenthesis 0 comma 30 right parenthesis; left parenthesis 2 comma 86 right parenthesis; and left parenthesis 4 comma 14 right parenthesis. There is shading below the curve.**
This description matches the features of the quadratic and its characteristics, specifically emphasizing that it correctly represents the physical situation described.