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# A plane lamina has an area $2{m^2}$at 10$^ \circ C$ then what is its area at 110$^ \circ C$? It's coefficient of superficial expansion is $2 \times {10^{ - 5}}/C$.A) 2.002${m^2}$B) 2.004${m^2}$C) 2.02${m^2}$D) 2.04${m^2}$

Last updated date: 09th Sep 2024
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Hint: According to the question we have to name the initial and final area, and find the appropriate temperature by subtraction. Substitute these given values in the formula by some simple steps we can get to the final answer.

Formula used:
$A = {A_0}\left( {1 + \beta \Delta T} \right)$

Complete Step by step solution:
Let A be the final area and and ${A_0}$ be initial area
According to question we have to write given values
A = $2{m^2}$
$\beta$ = $2 \times {10^{ - 5}}/C$

Formula is $A = {A_0}\left( {1 + \beta \Delta T} \right)$
A is the final area at the temperature 110$^ \circ C$
${A_0}$ is the initial area at 10$^ \circ C$
$\beta$ is the coefficient of superficial expansion
$\Delta T$ is the change in temperature

Calculate change in temperature
\eqalign{ & \Rightarrow \Delta T = {T_2} - {T_1} \cr & \Rightarrow \Delta T = 110 - 10 \cr & \therefore \Delta T = {100^ \circ }C \cr}

Substitute these values in the formula
$A = {A_0}\left( {1 + \beta \Delta T} \right)$
\eqalign{ & \Rightarrow A = 2\left( {1 + 2 \times {{10}^{ - 5}} \times 100} \right) \cr & \Rightarrow A = 2\left( {1 + 0.002} \right) \cr & \Rightarrow A = 2\left( {1.002} \right) \cr & \therefore A = 2.004{m^2} \cr}

Hence, the correct option is B.

We know that the coefficient of superficial expansion is defined as the increase in area per unit original area at 0$^ \circ C$ per unit rise in temperature.
We can also solve by taking a little bit of change in the formula, instead of A we use the differential form $\partial A$. It becomes A -${A_0}$ in the same way we can get the final answer. The student must remember we have to find an appropriate temperature.