A company sells cocoa powder in a cylindrical container that is 5 inches long with a circular base of diameter 4 inches. The company is planning to sell a family-size product that will use similar packaging scaled by a factor of 2.

The circumference of the original container is
?
inches. This is
?
inches less than the circumference of the new container.

To solve the problem, we need to calculate the circumference of both the original and the scaled-up cylindrical containers. 

First, let's find the circumference of the original container.

The formula for the circumference \(C\) of a circle is:
\[ C = \pi d \]
where \(d\) is the diameter.

For the original container:
\[ d = 4 \, \text{inches} \]
\[ C_{\text{original}} = \pi \times 4 \]

Next, let's compute the circumference:
\[ C_{\text{original}} = 4\pi \]
\[ C_{\text{original}} \approx 4 \times 3.14159 \approx 12.566 \, \text{inches} \]

Now, let's consider the scaled-up container. The scaling factor is 2, so the diameter of the new container will be:
\[ d_{\text{new}} = 2 \times 4 = 8 \, \text{inches} \]

The circumference of the new container will be:
\[ C_{\text{new}} = \pi \times 8 \]

Next, let's compute the circumference:
\[ C_{\text{new}} = 8\pi \]
\[ C_{\text{new}} \approx 8 \times 3.14159 \approx 25.133 \, \text{inches} \]

To find how much more the circumference of the new container is compared to the original container, we subtract the circumference of the original container from the circumference of the new container:
\[ \text{Difference} = C_{\text{new}} - C_{\text{original}} \]
\[ \text{Difference} = 8\pi - 4\pi \]
\[ \text{Difference} = 4\pi \]
\[ \text{Difference} \approx 4 \times 3.14159 \approx 12.566 \, \text{inches} \]

Therefore, the circumference of the original container is approximately 12.566 inches, which is approximately 12.566 inches less than the circumference of the new container.

To summarize:
1. The circumference of the original container is approximately \(12.566\) inches.
2. This is approximately \(12.566\) inches less than the circumference of the new container.

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:
1. What is the volume of the original container?
2. How does the volume of the new container compare to the original container?
3. What is the surface area of the original container?
4. How much larger is the surface area of the new container compared to the original?
5. What is the radius of the original container's base?
6. If the height of the container also scales by a factor of 2, what will be the height of the new container?
7. How would the circumference change if the scaling factor was 3 instead of 2?
8. If the original container is filled with cocoa powder, how many times more cocoa powder can the new container hold?

Tip: When scaling geometric shapes, remember that linear dimensions (like radius and height) scale directly by the factor, areas scale by the square of the factor, and volumes scale by the cube of the factor.

Explore how to calculate the circumference of cylindrical containers, comparing an original container with a scaled-up version. Learn the mathematical formula for circumference, apply scaling principles in geometry, and understand the difference in circumferences between scaled objects. Suitable for high school students studying geometry and scaling concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation