The jar below is approximately a cylinder with a height of 18 cm and a radius of 7 cm. The jar is completely filled to the top with jelly beans.
The jar below is approximately a cylinder with a height of 18 cm and a radius of 7 cm. The jar is completely filled to the top with jelly beans.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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*Challenge Question:*
What would be a good lower bound for the number of jelly beans in the jar and why?
**Explanation:**
To estimate the number of jelly beans, calculate the volume of the cylinder:
1. **Volume of the Cylinder:**
\[ V = \pi r^2 h \]
\[ V = \pi (7)^2 (18) \]
\[ V = \pi (49)(18) \]
\[ V \approx 2772 \pi \, \text{cubic centimeters} \]
2. **Volume of an Average Jelly Bean:**
Jelly beans are irregular, but roughly assuming each bean occupies about 2 cm³ (taking into account the empty spaces).
3. **Estimate of Jelly Beans:**
\[ \frac{\text{Total Volume of Jar}}{\text{Volume per Jelly Bean}} \approx \frac{2772 \pi}{2} \]
\[ \approx \frac{8716.1}{2} \]
\[ \approx 4358 \]
**Conclusion:**
A good lower bound estimate would be approximately 4300 jelly beans, considering packing efficiency and empty spaces between the beans.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fe4e0d9-a441-4076-9243-8e98c4a90bfe%2F18495b69-ce78-49b1-87a6-7033aa4106cb%2Fadp8hon_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Estimating the Number of Jelly Beans in a Cylindrical Jar**
The jar below is approximately a cylinder with a height of 18 cm and a radius of 7 cm. The jar is completely filled to the top with jelly beans.
*Challenge Question:*
What would be a good lower bound for the number of jelly beans in the jar and why?
**Explanation:**
To estimate the number of jelly beans, calculate the volume of the cylinder:
1. **Volume of the Cylinder:**
\[ V = \pi r^2 h \]
\[ V = \pi (7)^2 (18) \]
\[ V = \pi (49)(18) \]
\[ V \approx 2772 \pi \, \text{cubic centimeters} \]
2. **Volume of an Average Jelly Bean:**
Jelly beans are irregular, but roughly assuming each bean occupies about 2 cm³ (taking into account the empty spaces).
3. **Estimate of Jelly Beans:**
\[ \frac{\text{Total Volume of Jar}}{\text{Volume per Jelly Bean}} \approx \frac{2772 \pi}{2} \]
\[ \approx \frac{8716.1}{2} \]
\[ \approx 4358 \]
**Conclusion:**
A good lower bound estimate would be approximately 4300 jelly beans, considering packing efficiency and empty spaces between the beans.
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