Educational worksheet on the history and significance of Capitol Hill in Washington, D.C.

It looks like you've uploaded a text-based problem related to the Capitol Hill in Washington, D.C., involving geometry and proportions. Let's go through the problem step by step and solve it.

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Problem Summary:

The task involves designing a new capitol building with a circular central courtyard. The design is based on a scale model where:

- The diameter of the circular courtyard in the model is 10 cm.
- The actual diameter of the courtyard in the real building will be 100 meters.
- The height of the dome in the model is 5 cm.
- We are to find the actual height of the dome in the real building.

Additionally, there’s a second part:
> "When the builders are finished, they realize that the scale of the model was incorrect. The actual courtyard should be 120 meters in diameter. What would the new height of the dome be?"

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Step 1: Understand the Scale

We are given:

- Model courtyard diameter = 10 cm
- Actual courtyard diameter = 100 meters

Let’s convert units for consistency.

Convert 100 meters to centimeters:

> $ 100 \text{ m} = 100 \times 100 = 10,000 \text{ cm} $

So the scale of the model is:

$$
\frac{\text{Model}}{\text{Actual}} = \frac{10 \text{ cm}}{10,000 \text{ cm}} = \frac{1}{1000}
$$

So, the model is 1/1000th the size of the actual building.

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Step 2: Find the Actual Height of the Dome (Original Plan)

In the model, the dome height is 5 cm.

Using the scale factor:

$$
\text{Actual height} = 5 \text{ cm} \times 1000 = 5000 \text{ cm}
$$

Convert to meters:

$$
5000 \text{ cm} = 50 \text{ meters}
$$

✔ So, the original planned height of the dome is 50 meters.

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Step 3: Adjust for New Courtyard Diameter (120 meters)

Now, the builders realize the courtyard should be 120 meters, not 100 meters.

But the model was designed with a 10 cm diameter corresponding to 100 meters.

So the scale was based on:

$$
10 \text{ cm} : 100 \text{ m} \Rightarrow 1 \text{ cm} : 10 \text{ m}
$$

So each centimeter in the model represents 10 meters in reality.

But now, the actual courtyard is 120 meters.

To keep the same proportional design, we need to scale up the entire model proportionally.

Let’s find the new scale factor.

- Original model diameter: 10 cm → 100 m
- New actual diameter: 120 m

So the ratio of new to original actual size is:

$$
\frac{120}{100} = 1.2
$$

So the actual building is now 1.2 times larger than originally planned.

Therefore, all dimensions (including dome height) must be multiplied by 1.2.

Original dome height = 50 meters
New dome height = $ 50 \times 1.2 = 60 $ meters

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✔ Final Answers:

1. Original height of the dome: 50 meters
2. New height of the dome (for 120-meter courtyard): 60 meters

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🔍 Explanation:

- The model uses a scale of 1:1000 (1 cm = 1 meter? Wait — let’s double-check).

Wait! There’s a potential confusion here.

Earlier we said:

- 10 cm (model) → 100 m (real)
- So 1 cm → 10 m

That means the scale is 1 cm : 10 m, or 1:1000 in ratio (since 10 m = 1000 cm).

Yes, correct.

So, if the actual courtyard is now 120 m, then the model would need to have a diameter of:

$$
\frac{120 \text{ m}}{10 \text{ m/cm}} = 12 \text{ cm}
$$

So the model would be scaled up from 10 cm to 12 cm.

Then, the dome height in the model was 5 cm. Now, since the entire model is scaled by a factor of $ \frac{12}{10} = 1.2 $, the new model dome height is:

$$
5 \text{ cm} \times 1.2 = 6 \text{ cm}
$$

Then actual height = $ 6 \text{ cm} \times 1000 = 6000 \text{ cm} = 60 \text{ meters} $

Same result.

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✔ Final Answer:

- Original dome height: 50 meters
- Revised dome height (for 120-meter courtyard): 60 meters

This ensures the proportions of the building remain consistent with the original design.

Let me know if you'd like a diagram or further explanation!