Solve for #h#: #V=1/3 pi r^2 h#?

1 Answer
Geoff K.
Sep 7, 2017

Move terms that don't have #h# in them to the other side (using addition/subtraction), then move all factors other than #h# to the other side (using multiplication/division).

Explanation:

If we want to solve #V=1/3pir^2h# for #h#, we need to isolate the term with #h# (already done), and then multiply both sides by the inverses of everything other than #h#.

#color(white)(=>)Vcolor(white)(xx 3/(pir^2))=1/3pir^2h#

#=>Vcolor(red)(xx 3/(pir^2))=1/3pir^2hcolor(red)(xx 3/(pir^2))#

The multiplication by #3/(pir^2)# to both sides is our choice; we do this so that each piece other than #h# has a multiplicative inverse that cancels it off.

#=>Vxx 3/(pir^2)=1/color(orange)cancelcolor(black)(3)color(magenta)cancelcolor(black)(pir^2)h xx color(orange)cancelcolor(black)3/color(magenta)cancelcolor(black)(pir^2)#

#=> color(white)(V xx)(3V)/(pir^2)=color(white)(1/cancel3 cancel(pir^2))h#

Thus, our cone volume formula, when solved for #h#, is

#h=(3V)/(pir^2)#

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