Relative rates of change.

[Rumor]

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!

 

[BigGlenntheHeavy]

\(\displaystyle Hint: \ V \ = \ \frac{4}{3}\pi r^{3}, \ \frac{dV}{dt} \ = \ 4\pi r^{2}\frac{dr}{dt}\)

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

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

\(\displaystyle Hence, \ \frac{dr}{dt} \ = \ \frac{400\mu m^{3}/day}{400\pi \mu m^{2}} \ = \ \frac{1}{\pi}\mu m/day.\)

\(\displaystyle Note: \ I've \ edited \ the \ above \ per \ Subhotosh \ Khan, \ I \ stand \ corrected.\)

 

[BigGlenntheHeavy]

\(\displaystyle Afterthought: \ 1 \ micrometer \ = \ 1\mu m \ = \ 10^{-6}m\)

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

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

 

[BigGlenntheHeavy]

Another afterthought:

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

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

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

\(\displaystyle Amazing, \ isn't \ it? \ For \ you \ young \ men \ and \ women, \ think \ of \ all \ the \ things \ yet \ to \ be \ discovered.\)

 

[Deleted member 4993]

(dr/dt) should be expressed with a unit (dimension) - µm/day.