Question

In a steel plate of length \(100\) cm and breadth \(50\) cm, there is a hole in the shape of an equilateral triangle of side \(10\) cm. The coefficient of linear expansion of steel is \(1.2\times10^{-5}\,{}^{\circ}\mathrm{C}^{-1}\). The temperature of the steel plate is increased by \(100^\circ\mathrm{C}\). What will be the percentage change in the area of the triangular hole?

• A. \(0.24\%\)
• B. \(-0.24\%\)
• C. \(0.12\%\)
• D. \(-1.2\%\)

Hint:

A hole expands just like the surrounding material. For area expansion, \[ \beta=2\alpha. \] Percentage change: \[ \beta\Delta T\times100. \]

Answer 1

The Correct Option is A

Concept: A hole in a metal sheet expands exactly as if it were filled with the same material. Coefficient of superficial expansion: \[ \beta=2\alpha. \] Percentage increase in area: \[ \frac{\Delta A}{A}\times100 = \beta\Delta T\times100. \]

Step 1: Calculate the superficial expansion coefficient. \[ \beta=2\alpha \] \[ =2(1.2\times10^{-5}) \] \[ =2.4\times10^{-5}\ ^\circ\mathrm{C}^{-1}. \]

Step 2: Calculate the fractional increase in area. \[ \frac{\Delta A}{A} = \beta\Delta T \] \[ = (2.4\times10^{-5})(100) \] \[ = 2.4\times10^{-3}. \]

Step 3: Convert into percentage. \[ 2.4\times10^{-3}\times100 = 0.24\%. \] \[\begin{aligned} \boxed{0.24\%} \end{aligned}\] Hence, option \(\mathbf{(A)}\) is correct.