Question 1128630

Let {{{x}}} be the speed (rpm) 
Let {{{y}}} be the diameter of the shaft (inches) 
Let {{{z}}} be the horsepower transmitted 

Then the first sentence of the problem is expressed in terms of {{{x}}}, {{{y}}}, and {{{z}}} as: 


{{{z = kx/y^3}}}  for some constant {{{k}}} 

You are told that when  {{{y = 3}}} and {{{x = 110}}}, then {{{z = 40}}}: 

{{{40 = k(110)/(3)^3 = (110/27)k }}}

{{{40/(110/27) = k }}}

 {{{k =(40*27)/110}}}

{{{k =108/11}}}

=>

{{{z = kx/y^3}}}

{{{z = ((108/11)x)/y^3) }}}

{{{z = 108x/11y^3 }}}


now, what diameter must the shaft have in order to safely transmit  {{{60}}} hp at {{{150}}} rpm

You are given a value for {{{x=150}}} and {{{z=60}}}, and are asked the value of {{{y}}}: 

{{{z = 108x/11y^3 }}}...plug in given values

{{{60= (108*150)/(11y^3) }}}

{{{y^3 = (108*150)/(11*60)}}}

{{{y^3 = 270/11}}}

{{{y = root(3,270/11)}}}

{{{y = root(3,270)/ root(3,11)}}}

{{{y = 6.46330/ 2.22398}}}

{{{y = 2.91}}} (inches)