The surface area of a metal plate is $2.4\times {{10}^{-2}}{{m}^{2}}$ at ${{20}^{0}}C$ when the plate is heated to ${{185}^{0}}C$, its area increases by $0.8c{{m}^{2}}$. Find the coefficient of areal expansion of the metal.

Hint: This problem can be solved by using the direct formula for the change in surface area of a body when it is heated, in terms of the temperature change of the body, the initial area of the surface and the coefficient of areal expansion of the material.

Formula used: $\Delta A=A\beta \Delta T$

Complete step by step answer:
We will use the formula for the change in surface area of a body when subjected to a temperature change. Hence, let us write the formula.
The change $\Delta A$ in surface area of a body when it is subjected to a temperature change $\Delta T$ is given by
$\Delta A=A\beta \Delta T$ --(1)
where $A$ is the initial surface area of the body and $\beta $ is the coefficient of areal expansion of the material.
Now, let us analyze the question.
The initial surface area of the metal plate is $A=2.4\times {{10}^{-2}}{{m}^{2}}$
The initial temperature of the body is ${{T}_{i}}={{20}^{0}}C=20+273=293K$ $\left( \because {{T}^{0}}C=\left( T+273 \right)K \right)$
The final temperature of the body is ${{T}_{f}}={{185}^{0}}C=185+273=458K$ $\left( \because {{T}^{0}}C=\left( T+273 \right)K \right)$
Therefore, the change in temperature of the body is $\Delta T={{T}_{f}}-{{T}_{i}}=458-293=165K$
The change in area of the metal plate is $\Delta A=0.8c{{m}^{2}}=0.8\times {{10}^{-4}}{{m}^{2}}$ $\left( \because 1c{{m}^{2}}={{10}^{-4}}{{m}^{2}} \right)$
Let the coefficient of area expansion of the material of the metal plate be $\beta $.
Now, using (1), we get,
$0.8\times {{10}^{-4}}=2.4\times {{10}^{-2}}\times \beta \times 165$
$\therefore \beta =\dfrac{0.8\times {{10}^{-4}}}{2.4\times {{10}^{-2}}\times 165}=2.02\times {{10}^{-5}}{{K}^{-1}}$

Therefore, the coefficient of area expansion of the material is $2.02\times {{10}^{-5}}{{K}^{-1}}$.

Note: Students must note that it has been observed that usually the coefficient of areal expansion is twice that of the coefficient of linear expansion and the coefficient of volumetric expansion is three times the coefficient of linear expansion. Hence, if we had been given the coefficient of linear expansion, we could have found out the coefficient of areal expansion by using this observation.
Students must also note that the temperature change in the Celsius scale and the Kelvin scale both are the same but it is always a good practice to convert the temperatures in the Kelvin scale for thermodynamics problems even when it is not required in the final answer.