Educational worksheet titled "A Time Capsule From 1883" featuring a time capsule image and historical facts.

It looks like you've uploaded a text-based image of a problem titled "A Time Capsule From 1883", which describes a historical mathematical puzzle involving a cylindrical time capsule. Let's go through the problem step by step and solve it.

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🔍 Problem Summary:

- A time capsule was buried in 1883 with a cylinder-shaped container.
- The capsule is 10 inches high and has a diameter of 12 inches.
- It was sealed with a cylindrical lid that fits snugly on top.
- The lid has a height of 1 inch and the same diameter (12 inches).
- The capsule is made of cast iron, and its density is 7.2 g/cm³.
- The weight of the capsule is given as 156 pounds.
- The question: Is this weight reasonable?

We are to determine whether the stated weight of 156 pounds is plausible based on the dimensions and material density.

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✔ Step-by-Step Solution

#### Step 1: Understand the Geometry

The capsule consists of:
- A main cylinder: height = 10 in, diameter = 12 in → radius = 6 in
- A lid: height = 1 in, same diameter → radius = 6 in

So total volume = volume of main cylinder + volume of lid

But note: Since the lid is part of the capsule, we assume both are solid cast iron.

> We will compute the total volume of the capsule (both parts), convert to cubic centimeters, then use density to find mass in grams, then convert to pounds.

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#### Step 2: Volume of a Cylinder

$$
V = \pi r^2 h
$$

Where:
- $ r $ = radius
- $ h $ = height

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##### Main Cylinder:
- $ r = 6 $ in
- $ h = 10 $ in

$$
V_{\text{main}} = \pi (6)^2 (10) = \pi \cdot 36 \cdot 10 = 360\pi \text{ in}^3
$$

##### Lid:
- $ r = 6 $ in
- $ h = 1 $ in

$$
V_{\text{lid}} = \pi (6)^2 (1) = 36\pi \text{ in}^3
$$

##### Total Volume:
$$
V_{\text{total}} = 360\pi + 36\pi = 396\pi \text{ in}^3
$$

Use $ \pi \approx 3.1416 $

$$
V_{\text{total}} \approx 396 \times 3.1416 \approx 1246.08 \text{ in}^3
$$

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#### Step 3: Convert Cubic Inches to Cubic Centimeters

We know:
$$
1 \text{ in} = 2.54 \text{ cm}
\Rightarrow 1 \text{ in}^3 = (2.54)^3 \text{ cm}^3 = 16.387 \text{ cm}^3
$$

So:
$$
V_{\text{cm}^3} = 1246.08 \times 16.387 \approx ?
$$

Let’s calculate:

$$
1246.08 \times 16.387 \approx ?
$$

Break it down:
- $ 1200 \times 16.387 = 19,664.4 $
- $ 46.08 \times 16.387 \approx 754.6 $
- Total ≈ $ 19,664.4 + 754.6 = 20,419 \text{ cm}^3 $

So:
$$
V \approx 20,419 \text{ cm}^3
$$

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#### Step 4: Use Density to Find Mass

Density of cast iron = 7.2 g/cm³

$$
\text{Mass} = \text{Volume} \times \text{Density} = 20,419 \times 7.2
$$

Calculate:
$$
20,000 \times 7.2 = 144,000 \\
419 \times 7.2 = 2,992.8 \\
\text{Total} \approx 146,992.8 \text{ grams}
$$

Convert to kilograms:
$$
146,992.8 \div 1000 = 146.99 \text{ kg}
$$

Now convert to pounds:
$$
1 \text{ kg} \approx 2.2046 \text{ lbs}
\Rightarrow 146.99 \times 2.2046 \approx ?
$$

Calculate:
- $ 147 \times 2.2046 \approx 324.0 $
- But we have 146.99 → slightly less
- $ 146.99 \times 2.2046 \approx 323.8 \text{ lbs} $

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#### Step 5: Compare with Given Weight

- Calculated weight: ≈ 324 pounds
- Given weight: 156 pounds

This is almost double the claimed weight.

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✘ Conclusion:

The claimed weight of 156 pounds is not reasonable.

If the capsule were made entirely of solid cast iron with the given dimensions, it should weigh about 324 pounds, not 156.

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🤔 Possible Explanations:

1. The capsule is not solid — perhaps it's hollow or only the shell is iron.
2. The lid is not solid — maybe it's just a cover.
3. Material is not pure cast iron — perhaps thinner walls or lighter alloy.
4. Measurement error — maybe the dimensions are approximate.

But assuming the description says "cast iron" and full dimensions, the weight seems too low.

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✔ Final Answer:

No, the weight of 156 pounds is not reasonable.
Based on the dimensions and the density of cast iron, the capsule should weigh approximately 324 pounds, not 156. Therefore, either the capsule is not solid, or there's an error in the reported weight.

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Let me know if you'd like to explore how thick the walls could be for a 156-pound capsule!