A liquid having a coefficient of volume expansion γ is filled in a...

Let us assume that the volume of the liquid in the vessel is ${V}_{0} and the area of cross-section of the vessel is${A}_{0}$, then</p><p>${l}_{0} = \dfrac{{V}_{0}}{{A}_{0}}$</p><p>Similarly,</p><p>$l = \dfrac{V}{A} = \dfrac{{V}_{0}\left(1 + \gamma ∆ T\right)}{{A}_{0}\left(1 + 2\alpha ∆ T\right)}$</p><p>$⇒ l = {l}_{0}\left(1 + \gamma ∆ T\right){\left(1 + 2\alpha ∆ T\right)}^{- 1}$</p><p>Using binomial approximation, we get</p><p>$l = {l}_{0}\left(1 + \gamma ∆ T\right)\left(1 - 2\alpha ∆ T\right)$</p><p>$⇒ l = {l}_{0}\left[1 + \left(\gamma - 2\alpha\right) ∆ T - 2\alpha\gamma{\left(∆ T\right)}^{2}\right]$</p><p>Neglecting$2\alpha\gamma{\left(∆ T\right)}^{2}$, since it is insignificant as compared to \left(\gamma - 2\alpha\right) ∆ T$, we get

$l \approx {l}_{0}\left[1 + \left(\gamma - 2\alpha\right) ∆ T\right] ⇒ l \approx {l}_{0}\left[1 + \left(5\alpha - 2\alpha\right) ∆ T\right] ⇒ l \approx {l}_{0}\left[1 + 3\alpha ∆ T\right]$

So, $n = 3$