1. Given data

• Initial diameter of gold ring: $D_0 = 6.230\ \text{cm}$
• Diameter of wooden bangle: $D = 6.241\ \text{cm}$
• Initial temperature: $T_0 = 27^\circ \text{C}$
• Coefficient of linear expansion of gold: $\alpha = 1.4 \times 10^{-5}\ \text{K}^{-1}$

We need the ring diameter to increase from $6.230$ cm to $6.241$ cm.

2. Use linear expansion for diameter

For a ring, the diameter expands just like any linear dimension: $D = D_0(1 + \alpha \Delta T)$

So, $6.241 = 6.230(1 + \alpha \Delta T)$

3. Solve for $\Delta T$

$\frac{6.241 - 6.230}{6.230} = \alpha \Delta T$

$\frac{0.011}{6.230} = 1.4 \times 10^{-5} \Delta T$

First compute: $\frac{0.011}{6.230} \approx 0.00176565$

Now, $\Delta T = \frac{0.00176565}{1.4 \times 10^{-5}}$

$\Delta T \approx 126.12\ \text{K}$

4. Find the required final temperature

$T = T_0 + \Delta T$

$T = 27 + 126.12$

$T \approx 153.12^\circ \text{C}$

5. Match with options

Closest option is: $\boxed{152.7^\circ \text{C}}$

So the correct option is D.

6. Verification with stored answer

Stored correct answer: D

Our derived answer: D

They match.