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 time capsule buried in 1883. The task is to determine the
original volume of a cylindrical time capsule based on information about its dimensions and how it was modified.
Let’s carefully analyze the problem and solve it step by step.
---
🔍 Problem Summary:
- A cylindrical time capsule was buried in 1883.
- It had a
height of 40 inches and a
diameter of 20 inches.
- After being unearthed, the capsule was
cut into two equal parts (likely horizontally), resulting in
two smaller cylinders.
- Each half has:
- Height = 20 inches (half of original)
- Diameter = 20 inches (unchanged)
- The
volume of each half is given as
6283.19 cubic inches.
- We are to find the
original volume of the full cylinder
before it was cut.
---
🧮 Step-by-Step Solution:
We know the formula for the volume of a cylinder:
\[
V = \pi r^2 h
\]
Where:
- \( r \) = radius
- \( h \) = height
#### Step 1: Find the radius
Given diameter = 20 inches →
\[
r = \frac{20}{2} = 10 \text{ inches}
\]
#### Step 2: Use the volume of one half
Each half has:
- Radius = 10 inches
- Height = 20 inches
- Volume = 6283.19 in³
Let’s verify this volume using the formula:
\[
V_{\text{half}} = \pi r^2 h = \pi (10)^2 (20) = \pi \cdot 100 \cdot 20 = 2000\pi
\]
Now compute numerical value:
\[
2000\pi \approx 2000 \times 3.14159 = 6283.18 \text{ in}^3
\]
This matches the given volume (6283.19 in³ — slight rounding difference).
So,
each half has volume ≈ \( 2000\pi \) in³.
#### Step 3: Original volume
The full cylinder would be twice the volume of one half:
\[
V_{\text{original}} = 2 \times 6283.19 = 12,566.38 \text{ in}^3
\]
Alternatively, calculate directly:
Original height = 40 inches
\[
V = \pi r^2 h = \pi (10)^2 (40) = \pi \cdot 100 \cdot 40 = 4000\pi
\]
\[
4000\pi \approx 4000 \times 3.14159 = 12,566.36 \text{ in}^3
\]
---
✔ Final Answer:
\[
\boxed{12,566.36} \text{ cubic inches}
\]
This is the
original volume of the time capsule before it was cut.
---
💡 Additional Insight:
The problem likely uses this scenario to illustrate how volumes scale with dimensions. Even though the height was halved, the volume was also halved — consistent with the formula. The key idea is that
volume scales linearly with height when radius is constant.
Also, the mention of "1883" and the use of π suggests a historical context where precise measurement and geometry were important — possibly tied to engineering or preservation efforts of the era.
Let me know if you'd like a diagram or further explanation!
Parent Tip: Review the logic above to help your child master the concept of 7th grade reading comprehension worksheet with answers.
