Question

Please help!

Answer 1

Use differentials to find the change in mass 'm', as a percentage, when the radius 'r' of a solid sphere increases by 2%. Note that the mass of a sphere of uniform density is given by the following equation:
\[m = \frac{ 4}{ 3 }\rho \pi r ^{3}\]

Answer 2

do you know how to take the differential?

Answer 3

I got that to be\[\frac{ dm }{ dr }= 4\rho \pi r ^{2}\]
is this right and what should I do next please? thank you

Answer 4

so that is the derivative dm/dr. Cross multiply to move the dr to the right side please.

Answer 5

ok so that would be\[dm= 4\rho \pi r ^{2} dr \]
Would i then take dr to be 1.02 please?

Answer 6

know you need to find the change in mass as a percentage ...agreed?

Answer 7

yes, that makes sense

Answer 8

ok so form a ratio of dm to m and see what you get

Answer 9

you now have expressions for both dm and m

Answer 10

I've got \[\frac{ dm }{ m }=3 \pi dr \] but I think I might have made a mistake...

Answer 11

yes, try it again.
hint: the final expression will be in terms of dm, m, r and dr and a constant

Answer 12

I think this is right now!
\[\frac{ dm }{ m }=\frac{ 3 }{ r }dr\]

Answer 13

Excellent! Can you complete the problem now?

Answer 14

I think so, would I take dr to be 1.02r now please?

Answer 15

dr = .02 , r =1

Answer 16

Ah, that makes more sense! Thank you so much for taking the time to go through all of this with me I really really appreciate it!

Answer 17

you are very welcome...excellent work...have a great New Year!

Answer 18

you too!