How to Calculate Volume Thermal Expansion Coefficient?

[phrygian]

Homework Statement

Prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions: B = Ax + Ay + Az

Homework Equations

B = (dV/V)/dT
A = (dL/L)/dT

The Attempt at a Solution

My thought was using dV/V = ((dLx*dLy*dLz)/(Lx*Ly*Lz)) but when you use this it is clear that dL^3/L^3 does not equal 3* dL/L

What am I doing wrong?

Thanks for the help

 

[rl.bhat]

dV= ( Lx +dLx)( Ly +dLy)( Lz +dLz) - Lx*Ly*Lz
Simplify this and proceed.

 

[phrygian]

Can you explain mathematically how you got there?

Also, the problem statement says at the end (So for an isotropic solid, which expands the same in all directions, B = 3A) if that makes the problem simpler.

 

[rl.bhat]

Lx, Ly and Lz are the lengths of the block. When the temperature is raised through 1 degree C, the new lengths will be Lx + Ax, Ly + Ay and Lz+ Az.
New volume will be (Lx + Ax)( Ly + Ay)( Lz+ Az). Original volume is Lx*Ly*Lz.
Sp dV = (Lx + Ax)( Ly + Ay)( Lz+ Az) - Lx*Ly*Lz
Find dV/V. Neglect the terms like Ax*Ay and so on because they are very small quantities.

 

[Redbelly98]

dV= ( Lx +dLx)( Ly +dLy)( Lz +dLz) - Lx*Ly*Lz
Simplify this and proceed.

Can you explain mathematically how you got there?

dV = (volume after expansion) - (initial volume)​

After expansion, the box dimensions have increased by dLx, dLy, and dLz, from their initial lengths Lx, Ly, and Lz.

EDIT:
rl responded faster than I.