Student Activity: To investigate the rate of change of height with respect to volume Use in connection with the interactive file, ‘Height with respect to volume’, on the student’s CD. 1. a. Using a calculator complete the following table for a container of radius r = 2 cm and height h: volume Volume (cm)3 h= π(r)2 (cm) 10 20 30 40 50 b. Draw a graph of the data represented in the table. © Project Maths Development Team 2011 Height with respect to volume Page 1 of 5 c. Did your data form a linear, quadratic or exponential graph? Explain your answer. ___________________________________________________________________________ ___________________________________________________________________________ d. Is the height of this cylinder proportional to its volume? Explain your answer. __________________________________________________________________________ __________________________________________________________________________ e. What does the slope mean in the context of this problem? __________________________________________________________________________ __________________________________________________________________________ f. Is the rate of change of the height with respect to volume increasing, decreasing, or constant for this problem? Explain your answer. __________________________________________________________________________ __________________________________________________________________________ 2. a. Using a calculator, complete the following table for a container of radius r = 3 cm and height h: volume Volume (cm)3 h= π(r)2 (cm) 10 20 30 40 50 © Project Maths Development Team 2011 Height with respect to volume Page 2 of 5 b. Draw a graph of the data represented in the table. c. Did your data form a linear, quadratic or exponential graph? Explain your answer. __________________________________________________________________________ __________________________________________________________________________ d. Is the height of this container proportional to its volume? Explain your answer. __________________________________________________________________________ __________________________________________________________________________ e. What does the slope mean in the context of this problem? __________________________________________________________________________ __________________________________________________________________________ f. Is the rate of change of the height with respect to volume increasing, decreasing, or constant for this problem? Explain your answer. __________________________________________________________________________ __________________________________________________________________________ 3. Using the two graphs you have drawn and the interactive file, do you agree with the statement that “The greater the radius of the container the slower the rate of change of the height (cm) with respect to the volume (cm)3.” Explain your reasoning. __________________________________________________________________________ __________________________________________________________________________ © Project Maths Development Team 2011 Height with respect to volume Page 3 of 5 4. Following the logic obtained in the above questions can you explain why, if the radius of a container is getting smaller as in the flask opposite, the graph of volume (cm)3 vs. height (cm)will be shaped as follows: __________________________________________________________________________ __________________________________________________________________________ 5. Following the logic obtained in the above questions can you explain why if the radius of a container is getting larger as in the container opposite, the graph of volume (cm3) vs. height (cm) will be shaped as follows: __________________________________________________________________________ __________________________________________________________________________ 6. Draw a rough sketch of the graph of the volume vs. the height of the following container. Note water cannot enter the handles. © Project Maths Development Team 2011 Height with respect to volume Page 4 of 5 7. Draw a rough sketch of the shape of the container represented by the following graph as water is being poured into it. 8. Draw a rough sketch of the shape of the container represented by the following graph as water is being poured into it. © Project Maths Development Team 2011 Height with respect to volume Page 5 of 5

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