The rectangular plate shown if Fig. 1.7 has an area `A_i`. If the temperature increases by `DeltaT`, each dimension increases according to `DeltaL=alphaLDeltaT`, where `alpha` is the average coefficient of linear expansion. Show that the increase in area is `DeltaA=2alphaA_iDeltaT`. What approzimation does this expansion assume?

Explanation

We expect the area to increase in thermal expansion. It is neat that the coefficent of area expansion is just twice the coefficient of linear expansion. We will use the definitions of coefficients of linear and area expansion.
From the diagram is Fig. 1.8, we see that the change in area is
`DeltaV=lDeltaw+wDeltal+DeltawDeltal`
Since `Deltal` and `Deltaw` are each small quantities, the product `DeltawDeltal` will be very small as compared to the original or final area.
Therefore, we assume `DeltawDeltalprop0`
Since `Deltaw=walphaDeltaT` and `Deltal=lalphaDeltaT`
We then have `DeltaA=lwalphaT+wlalphaDeltaT`
Finally, since `A=lw`, we have `DeltaA=2alphaADeltaT`