Math Problem Statement

A hot air balloon is rising at a constant rate. The relationship between its height above the ground and the time since it left the ground is shown in the following graph. A scatter plot showing the relationship between time and height. The x-axis is labeled 'Time open parenthesis minutes close parenthesis' and ranges from 1 to 5 in increments of 1, while the y-axis is labeled 'Height open parenthesis meters close parenthesis' and ranges from 20 to 120 in increments of 20. One filled point is plotted, located at (2,40). Origin is highlighted by a filled point. An arrow starts from the origin (0, 0) and passes through the point (2,40) and rises in top right direction. [\small{1}] [\small{2}] [\small{3}] [\small{4}] [\small{5}] [\small{20}] [\small{40}] [\small{60}] [\small{80}] [\small{100}] [\small{120}] Which statements about the graph are true? Choose all answers that apply: Choose all answers that apply: (Choice A) The point [(0, 0)] shows that the hot air balloon is free to ride. A The point [(0, 0)] shows that the hot air balloon is free to ride. (Choice B) The point [(2, 40)] shows that the hot air balloon is at a height of [2 \text{ m}] after [40] seconds. B The point [(2, 40)] shows that the hot air balloon is at a height of [2 \text{ m}] after [40] seconds. (Choice C) None of the above C None of the above

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Solution

Let's analyze the statements based on the graph described:

  • Statement A: The point (0,0)(0, 0) shows that the hot air balloon is free to ride.

    • Analysis: The point (0,0)(0, 0) actually indicates that the balloon starts at ground level (0 meters) at the beginning of the observation (0 minutes). It doesn't suggest anything about the balloon being free to ride; it just marks the starting point.

  • Statement B: The point (2,40)(2, 40) shows that the hot air balloon is at a height of 2 meters after 40 seconds.

    • Analysis: This statement is incorrect. The point (2,40)(2, 40) on the graph represents the balloon being at a height of 40 meters after 2 minutes, not 2 meters after 40 seconds.

  • Statement C: None of the above.

    • Analysis: Since both Statement A and Statement B are incorrect, this statement is true.

Correct Answer: (Choice C) None of the above

Would you like any further details or explanations?

Here are five related questions to further explore this topic:

  1. How would the graph change if the balloon rose faster?
  2. What does the slope of the line in the graph represent?
  3. If the balloon starts descending at a constant rate, how would that be reflected in the graph?
  4. Can the height of the balloon ever decrease on this graph? Why or why not?
  5. What does a point like (3,60)(3, 60) represent on this graph?

Tip: When interpreting graphs, always pay close attention to the units and labels on the axes to ensure correct interpretation.

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Math Problem Analysis

Mathematical Concepts

Graph interpretation
Linear relationships

Formulas

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Theorems

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Suitable Grade Level

High school

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