To solve this problem, we need to account for the expansion of the metal tape due to the change in temperature. The tape gives a correct reading at 20℃, but the temperature has increased to 32℃. The coefficient of linear expansion $\alpha$ of the tape tells us how much the length of the tape changes with temperature.

Formula for Linear Expansion:

The formula for the change in length due to a temperature change is:

$\Delta L = L_0 \cdot \alpha \cdot \Delta T$

Where:

• $\Delta L$ is the change in length,
• $L_0$ is the original length,
• $\alpha$ is the coefficient of linear expansion,
• $\Delta T$ is the change in temperature.

Given:

• $L_0 = 30 \, \text{m}$ (measured length of the tape),
• $\alpha = 50 \times 10^{-6} / ^\circ \text{C}$,
• $\Delta T = 32^\circ \text{C} - 20^\circ \text{C} = 12^\circ \text{C}$.

Step 1: Calculate the Change in Length of the Tape

$\Delta L = 30 \cdot 50 \times 10^{-6} \cdot 12$

$\Delta L = 30 \cdot 600 \times 10^{-6}$

$\Delta L = 18 \times 10^{-3} = 0.018 \, \text{m}$

So, the tape has expanded by $0.018 \, \text{m}$ (or 1.8 cm).

Step 2: Calculate the Actual Distance Between the Lamp Posts

Since the tape expands, it gives a shorter measurement of the actual distance. Therefore, the actual distance $L_{\text{actual}}$ between the lamp posts is:

$L_{\text{actual}} = L_0 + \Delta L = 30 \, \text{m} + 0.018 \, \text{m} = 30.018 \, \text{m}$

Final Answer:

The actual distance between the lamp posts is 30.018 m.

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Would you like a detailed breakdown of any step, or do you have further questions?

Here are some related questions to explore:

1. How would the result change if the coefficient of linear expansion was smaller?
2. What would happen to the actual distance if the temperature increased further, say to 40°C?
3. What is the concept of coefficient of linear expansion, and how is it determined for different materials?
4. How does the expansion of materials affect construction and engineering designs?
5. How would the expansion of the tape be different if the material was made of a different substance, like steel instead of metal?

Tip: In problems like this, always keep track of the unit conversions to ensure consistency and avoid errors.