Click here to print a blank answer sheet.
Exploring the Size of Organisms
Surface:Volume Ratio
1. Calculate the surface area of a cube that is 1 cm X 1 cm X 1 cm. answer: ________ sq cm (or cm2) Calculate the volume of the cube in the example above. answer:_________cubic cm (or cm3) Calculate the surface to volume ratio (surface:volume). This can be done by dividing the surface area by the volume. answer:_________cm2/cm3 2. Consider a cube that is ten times bigger than the one in the example above.
Calculate the surface area of a cube that is 10 cm X 10 cm X 10 cm. answer: ________ cm2 Calculate the volume of this cube. answer:_________cm3 Calculate the surface to volume ratio. answer:_________cm2/cm3 3. When the cube was increased in size 10 times, how many times did the surface area increase? When the cube was increased in size 10 times, how many times did the volume increase?
4. As an object increases in size, it's surface area increases _____________ (faster or slower) than it's volume.
The concept described above has implications for cells because the plasma membrane forms the surface of the cell and is important in movement of materials into and out of the cell. Cells require a large amount of plasma membrane, and as a result, are limited in size. Larger organisms have solved the problem that limits cell size by being multicellular.
Large Organisms
In this example we will work with trees but the conclusions can be applied to any large organism.
Consider a tree that is 10 meters tall and a second tree that is 20 meters tall. Should the diameter of the larger trunk be twice as much as that of the small trunk? This question will be answered below.
5. Suppose that a tree that is 10 m tall and the diameter of it's trunk is 0.1 meters. Calculate the cross-sectional area of the trunk. (A =
r2) ______ sq m
6. Another tree is exactly as large as the tree discussed above. It is 20 meters tall and has a diameter of 0.2 meters. Calculate the cross-sectional area of the trunk. ______ sq m
7. Did the area of the trunk also double? What happened? Why?
The area of a circle is a function of the radius squared (A =
Volume is a function of length X width X height. Therefore, if the tree dimensions double, the volume increases by a factor of 23 = 8. As we can see from this calculation and the one above, volume increases more than the area of the trunk.
If the tree were three times as large, the volume (and thus weight) of the tree would be increased by 33 = 27 times but the cross-sectional area of the stem would be 32 = 9 times.
The strength of a tree trunk is proportional to its cross-sectional area; larger trunks are stronger. From above, we can see that tripling the size of a tree makes the trunk 9 times stronger but tree is 27 times heavier. What implications does this have for large organisms?
8. Hypothesize what trees can do to overcome this limitation on size.
Laboratory Exercise
Calculate the height and diameter of five trees of different sizes as described below.
Calculation of Height
The tangent of angle
in the triangle below is equal to A/B.
If we know the length of side B we can calculate the length of side A because A = tan
We can use the same principle to calculate the height of the tree in the diagram below.
The observer can use a tape measure to measure B in the diagram above. The angle can be calculated using clinometer. If this is not available, a protractor can be used. After A is calculated, the distance from the ground to the observer's eye must be added to A because the angle is measured by the observer at eye level.
Measurement of Diameter.
Diameter of trees is usually measured at approximately 4 feet above the ground (called DBH or Diameter at Breast Height). Diameter can be calculated by measuring the circumference of the stem and then converting this measurement to diameter using the formula C =
Data
9. After you measure the height of a tree, record it's diameter by first measuring its circumference and then converting that measurement to diameter using the formula D = C/
10. Record data from the rest of the class on your answer sheet.
11. Plot the class data. Put tree height on the X (horizontal) axis and diameter on the Y axis.
Conclusions
12. Does your data support your hypothesis? Why or why not?
13. Discuss how your findings might apply to animals. For example, elephants have very large, thick legs compared to the rest of their body. Are animals that have legs limited in how large they can be?
