Relative rates of change.

Rumor

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Sep 8, 2009
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Another problem that I'm stuck on: ( u = mu, since I don't know how to do the symbol)

"A spherical cell is growing at a constant rate of 400 um^3/day (1 um = 10^-6 m). At what rate is its radius increasing when the radius is 10 um?"

If someone could help me set this problem up, I'd appreciate it! Thanks!
 
Hint: V = 43πr3, dVdt = 4πr2drdt\displaystyle Hint: \ V \ = \ \frac{4}{3}\pi r^{3}, \ \frac{dV}{dt} \ = \ 4\pi r^{2}\frac{dr}{dt}

Given: dVdt = 400μm3/day, find drdt when r = 10μm.\displaystyle Given: \ \frac{dV}{dt} \ = \ 400 \mu m^{3}/day, \ find \ \frac{dr}{dt} \ when \ r \ = \ 10\mu m.

400μm3/day = 4π(100μm2)drdt\displaystyle 400\mu m^{3}/day \ = \ 4\pi(100\mu m^{2})\frac{dr}{dt}

Hence, drdt = 400μm3/day400πμm2 = 1πμm/day.\displaystyle Hence, \ \frac{dr}{dt} \ = \ \frac{400\mu m^{3}/day}{400\pi \mu m^{2}} \ = \ \frac{1}{\pi}\mu m/day.

Note: Ive edited the above per Subhotosh Khan, I stand corrected.\displaystyle Note: \ I've \ edited \ the \ above \ per \ Subhotosh \ Khan, \ I \ stand \ corrected.
 
Afterthought: 1 micrometer = 1μm = 106m\displaystyle Afterthought: \ 1 \ micrometer \ = \ 1\mu m \ = \ 10^{-6}m

1 nanometer = 1nm = 109m\displaystyle 1 \ nanometer \ = \ 1nm \ = \ 10^{-9}m

Thus, 1 μm = 103 nm. (Note that the micrometer is often called the micron.)\displaystyle Thus, \ 1 \ \mu m \ = \ 10^{3} \ nm. \ (Note \ that \ the \ micrometer \ is \ often \ called \ the \ micron.)
 
Another afterthought:

1 μm = 106 meters or 1 micron = 1 millionth of a meter.\displaystyle 1 \ \mu m \ = \ 10^{-6} \ meters \ or \ 1 \ micron \ = \ 1 \ millionth \ of \ a \ meter.

Now, to add insult to injury, 1 nm = 109 meters or 1 nm (nanometer) =1 billionth of a meter.\displaystyle Now, \ to \ add \ insult \ to \ injury, \ 1 \ nm \ = \ 10^{-9} \ meters \ or \ 1 \ nm \ (nanometer) \ =1 \ billionth \ of \ a \ meter.

Now, a meter is about a yard, think about it, ergo, how small is small and how big is big?\displaystyle Now, \ a \ meter \ is \ about \ a \ yard, \ think \ about \ it, \ ergo, \ how \ small \ is \ small \ and \ how \ big \ is \ big?

Amazing, isnt it? For you young men and women, think of all the things yet to be discovered.\displaystyle Amazing, \ isn't \ it? \ For \ you \ young \ men \ and \ women, \ think \ of \ all \ the \ things \ yet \ to \ be \ discovered.
 
(dr/dt) should be expressed with a unit (dimension) - µm/day.
 
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