tion:AnswerWe know that β is the coefficient of volumetric EXPANSION and α is the coefficient of linear expansion.Let us now consider a cube with dimensions l,b,h as the length breadth and height respectively. Let V be its volume. Let the initial temperature be T 1 . NEXT let's change the temperature from T 1 to T 2 . The cube will undergo expansion. Let the dimensions before expansion be l 1 ,b 1 ,h 1 ,V 1 and the dimensions after expansion be l 2 ,b 2 ,h 2 ,V 2 for length, breadth, height and volume respectively.From linear expansion theory we have,l 2 =l 1 (1+α△T),b 2 =b 1 (1+α△T) andh 2 =h 1 (1+α△T).The volume of the cube after expansion is GIVEN by,V 2 =l 2 ×b 2 ×h 2 Using the values of l 2 ,b 2 ,h 2 in the volume equation we have,V 2 =l 1 (1+α△T)×b 1 (1+α△T)×h 1 (1+α△T)⇒V 2 =(l 1 ×b 1 ×h 1 )(1+α△T) 3 Now the volume of the cube before expansion is given by V 1 =l 1 ×b 1 ×h 1 Using this,⇒V 2 =V 1 (1+α△T) 3 EXPANDING (1+α△T) 3 as (a+b) 3 =a 3 +b 3 +3a 2 b+3ab 2 we have,⇒V 2 =V 1 [1+3α△T+3(α△T) 2 +(α△T) 3 ]Now the value of 3(α△T) 2 +(α△T)≈0∴V 2 =V 1 (1+3α△T)⇒V 2 =V 1 +3V 1 α△Tor, V 2 −V 1 =3V 1 α△TNow V 2 −V 1 is nothing but the change in volume, △V.∴△V=3V 1 α△Tor, V 1 △T△V =3αor, β=3α as β= V 1 △T△V HENCE, proved.