1.) For a bacterial cell with a diameter of 1 um and a protist with a diameter 1
ID: 140269 • Letter: 1
Question
1.) For a bacterial cell with a diameter of 1 um and a protist with a diameter 15 um, what is the volume, surface area, and surface-to-volume (S:V) ratio for each cell assuming a perfect sphere? How would the surface-to-volume ratios change if you included the internal organelle membranes assuming it added 15 times the area of the plasma membrane?
2.) The brain weighs roughly 1 kg and contains 10^12 cells. Calculate the average volume of a brain cell assuming that each cell is filled entirely with water and 1 cubic centimeter of water weighs 1 g. What is the diameter of the average nerve assuming a perfect sphere?
Explanation / Answer
1) Diameter of bacterium = 1 um.
radius of bacterium = half the diameter = 0.5 um. = 0.5 x 10-6 m= 0.5 x10-4 cm
volume of sphere: r³= (4/3) (3.14) (0.5 x10-4 cm)3 = 1.33 x 3.14 x 0.125 x 10-12 cm3
= 0.523 x 10-12 cm3
surface area of sphere bacterium = 4r2= 4 x 3.14 x (0.5 x 10-4cm)2 = 4 x 3.14 x 0.25 x 10-8 cm2 = 3.14 x10-8 cm2.
Surface to volume ratio: Surface area / Volume = 3.14 x10-8 cm2/0.523 x 10-12 cm3= 6 x 104 cm-1.
New surface area : old surface area + 15 (old surface area) = 16 (old surface area) = 16 x 3.14 x10-8 cm2 = 50.24 x10-8 cm2.
New surface to volume ratio: 50.24 x10-8 cm2/0.523 x 10-12 cm3= 96.06 x 104 cm-1.
Diameter of protist = 15 um
radius of protist = 7.5 um
volume of sphere: r³= (4/3) (3.14) (7.5 x 10-4 cm)3 = 1761.83 x 10-12 cm3.
surface area of sphere: 4r2= 4 x 3.14 x (7.5 x 10-4)cm2= 706.5 x 10-8 cm2
Surface area to voume ratio for protist = 706.5 x 10-8 cm2 /1761.83 x 10-12 cm3 = 0.4010 x 104 cm-1.
if 15 times surface area is added, new surface area = old surface area +15 (old surface area) = 706.5 x 10-8 cm2 +15 (706.5 x 10-8 cm2 ) = 706.5 x 10-8 cm2 +10597.5 x10-8 cm2 = 11304 x10-8 cm2.
The new surface area to volume ratio: 11304 x10-8 cm2./ 1761.83 x 10-12 cm3 = 6.41 x 104 cm-1 .
2) weight of brain = 1 kg = 1000 g.
since 1 cm3 of water weighs 1 gram, 1000 cm3 volume of water will weigh 1000 grams, which is 1 kg.
therefore, total volume of a 1 kg brain is 1000 cm3.
number of cells = 1012
volume in one cell = total volume/ number of cells = 1000 cm3/ 1012= 10-9 cm3.
We know that volume is r³ and that the volume of one cell is 10-9 cm3
r³ = 10-9 cm3
r³ = (3/4) x 10-9 cm3
r³ = (3/4) x 10-9 cm3 x (1/3.14)
= 0.2388 x 10-9 cm3
After taking the cube root, r = 0.6204 x 10-3 cm.
diameter = 2r = 1.2408 x 10-3 cm.
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