An ice cream company sells their product in a cylindrical container that is \[16\text{ cm}\] tall and has a diameter of \[14\text{ cm}\]. They are considering switching to a new container that is a rectangular prism whose base is \[20 \times 13 \text{ cm}\].

The company wants the new container to have the same volume as the old container.

**How tall should the new container be?**
*Round to the nearest tenth.*

To find the height of the new rectangular container, we need to ensure that its volume matches the volume of the old cylindrical container.

### Step 1: Calculate the volume of the old cylindrical container.

The volume \( V \) of a cylinder is given by:

\[
V = \pi r^2 h
\]

Given:
- Diameter of the cylinder = \( 14 \text{ cm} \), so the radius \( r \) = \( \frac{14}{2} = 7 \text{ cm} \).
- Height of the cylinder \( h \) = \( 16 \text{ cm} \).

Substitute these values into the formula:

\[
V = \pi (7)^2 (16)
\]

\[
V = \pi \times 49 \times 16
\]

\[
V = 784\pi \text{ cubic centimeters}
\]

### Step 2: Set the volume of the new rectangular prism equal to the volume of the cylinder.

The volume \( V \) of a rectangular prism is given by:

\[
V = \text{length} \times \text{width} \times \text{height}
\]

Given:
- Length = \( 20 \text{ cm} \)
- Width = \( 13 \text{ cm} \)
- Height = \( h \) (which we need to find)

We know the volume of the new container must equal the volume of the old cylinder:

\[
20 \times 13 \times h = 784\pi
\]

\[
260h = 784\pi
\]

### Step 3: Solve for \( h \).

\[
h = \frac{784\pi}{260}
\]

\[
h \approx \frac{2463.01}{260} \text{ (using } \pi \approx 3.14159\text{)}
\]

\[
h \approx 9.47 \text{ cm}
\]

### Final Answer:
The height of the new container should be approximately \( 9.5 \text{ cm} \) when rounded to the nearest tenth.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How much more or less material is needed to produce the new container compared to the old one?
2. What is the surface area of the new rectangular container?
3. How does the height change if the base of the rectangular prism is increased?
4. What is the difference in height if the new container’s base were square instead?
5. How would the volume change if the height of the cylinder were increased by 25%?

**Tip:** When comparing volumes, always ensure you use the same units and remember to consider the shape-specific volume formulas for accurate calculations.

Learn how to calculate the height of a new rectangular container to match the volume of a cylindrical container. This problem involves geometry and volume calculations, applying formulas for cylinder and rectangular prism volumes. Suitable for students in Grades 7-9.

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