# Student Activity: To investigate the rate of change of height... respect to volume

```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.
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d. Is the height of this cylinder proportional to its volume? Explain your answer.
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e.
What does the slope mean in the context of this problem?
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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.
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© 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:
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__________________________________________________________________________
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
```