Let b = x, d = 2x. S = mg(1/3) / (4/4*J) for rectangle. I = bd^3 / 12. S = mg(1/3)*X/ (4*Y*bd^3). Case for (a) where b = 2x, d = x. I1 = mg(1/3) / (4*Y*2x^3).

Steps to Solve

1. Identify Variables Given:

   • Base ( b = x )
   • Depth ( d = 2x )

2. Write the Formula for the Moment of Inertia (I) The formula for the moment of inertia for a rectangular prism is given by: $$ I = \frac{bd^3}{12} $$ Substitute ( b ) and ( d ): $$ I = \frac{x(2x)^3}{12} = \frac{x(8x^3)}{12} $$

3. Simplify the Moment of Inertia (I) Simplifying the obtained formula gives: $$ I = \frac{8x^4}{12} = \frac{2x^4}{3} $$

4. Calculate Stress (S) From the problem, the formula for stress is: $$ S = \frac{mg(1/3) \cdot X}{(4 \cdot Y) \cdot I} $$ Substitute ( I ): $$ S = \frac{mg(1/3) \cdot X}{(4 \cdot Y) \cdot \left(\frac{2x^4}{3}\right)} $$

5. Simplify the Stress Equation Simplifying provides: $$ S = \frac{mg(1/3) \cdot X \cdot 3}{4Y \cdot 2x^4} = \frac{mgX}{8Yx^4} $$

6. Case for (a) For the case where ( b = 2x ) and ( d = x ) repeat the process: Substitute ( b ) and ( d ) into the moment of inertia: $$ I_1 = \frac{(2x)(x^3)}{12} = \frac{2x^4}{12} = \frac{x^4}{6} $$

7. Calculate ( S_1 ) for Case (a) Using the stress formula: $$ S_1 = \frac{mg(1/3)}{4Y \cdot (2x)(x^3)} $$ Simplifying gives: $$ S_1 = \frac{mg(1/3)}{8Yx^4} $$