Question

for a spherical object what is its mass in terms of the density of the material, its diameter and any necessary constants?

Answer 1

well volume of a sphere is \[\frac 43 \pi r^3\]

Answer 2

and then density = mass/volume

Answer 3

\[\rho=\frac mV\]
\[m=V\rho\]

Answer 4

so you have \[m = \frac 43\rho \pi r^3\]

Answer 5

where \(\rho\) = density

Answer 6

what about its diameter

Answer 7

change r to d/2

Answer 8

and \(r\) is the radius
(half the diameter)
\(r=\frac d2\)

Answer 9

radius = diameter/2 so just change r to d/2

Answer 10

ok so thats the final solution?

Answer 11

replace r with d/2....

Answer 12

ok cool and can i give you guys both metals or does just one of you get it?

Answer 13

hey so its (4/3)p(pi)(d/2)^3

Answer 14

right

Answer 15

is that right?

Answer 16

you can simplify the numbers a bit

Answer 17

unkle...it won't let me give you a metal now

Answer 18

why cant i give it to both of you

Answer 19

i dont mind about that

Answer 20

\[m=\frac43\rho\pi\left(\frac d2\right)^3\]
use
\[\left(\frac{a}{b}\right)^n=\frac {a^n} {b^n}\]

Answer 21

so it simplifies to 4p(pi)(d/2)?

Answer 22

i crossed out the 3's

Answer 23

what he meant was distribute the exponent

Answer 24

you can not cancel an index with denominator like that

Answer 25

oh so whats it turn to?

Answer 26

\[\left(\frac{a}{b}\right)^n=\frac {a^n} {b^n}\]
\[\left(\frac{d}{2}\right)^3=\frac {d^3} {2^3}\]

Answer 27

thanks :)

Answer 28

can you simplify the numbers in the equation now?

Answer 29

yes (4/3)p(pi)(d^3/8) ?

Answer 30

you can simplify it further

Answer 31

you can multiply the denominators

Answer 32

so 4p(pi)(d/8) ? ?

Answer 33

what happened?

Answer 34

can you simplify it?

Answer 35

you canceled the denominaotr and teh exponent???

Answer 36

it can't be simplified?

Answer 37

it can...you just simplified it wrong....multiply the denominators

Answer 38

don't touch anything else

Answer 39

so it (4/24)p(pi)(d^3) ?

Answer 40

thats better!

Answer 41

yes. now express 4/24 in lowest terms

Answer 42

ok so 1/6 p(pi)(d^3) ? thats final!!!!

Answer 43

right!!

Answer 44

your turn @UnkleRhaukus

Answer 45

\[\color{red}\checkmark\]

Answer 46

so the constants necessary to express the volume in terms of diameter are ;

the natural number; 6
the density of the material rho; \(\rho\)
the geometrical constant pi; \(\pi\)
the diameter of the sphere \(d\)
and the dimensional constant \(3\)