A can of soda pop has the shape of a right circular cylinder with an inside height of 6 inches and an inside diameter of 2 inches. When you pour the soda pop from the full can into a cylindrical glas with an inside diameter of 3 inches, about how many inches high is the soda pop in the glass?
(Note: The volume of a right circular cylinder is `pi r^(2) h`)

Options

A. `2 2/3`
B. `4`
C. `5`
D. `6 2/3`

Answer

To solve the problem, we need to find out how high the soda pop will be in the cylindrical glass after pouring it from the can. We will use the formula for the volume of a cylinder, which is given by:

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

where \( r \) is the radius and \( h \) is the height.

### Step 1: Calculate the volume of the soda can

1. **Find the radius of the soda can:**
- The diameter of the soda can is 2 inches, so the radius \( r \) is:
\[ r = \frac{\text{diameter}}{2} = \frac{2}{2} = 1 \text{ inch} \]

2. **Use the height of the soda can:**
- The height \( h \) of the soda can is given as 6 inches.

3. **Calculate the volume of the soda can:**
- Using the volume formula:
\[ V_{\text{can}} = \pi r^2 h = \pi (1)^2 (6) = 6\pi \text{ cubic inches} \]

### Step 2: Calculate the volume of the glass

1. **Find the radius of the glass:**
- The diameter of the glass is 3 inches, so the radius \( r \) is:
\[ r = \frac{\text{diameter}}{2} = \frac{3}{2} = 1.5 \text{ inches} \]

2. **Let the height of the soda in the glass be \( h \).**

3. **Write the volume of the glass:**
- The volume of the glass can be expressed as:
\[ V_{\text{glass}} = \pi r^2 h = \pi (1.5)^2 h = \pi (2.25) h = 2.25\pi h \text{ cubic inches} \]

### Step 3: Set the volumes equal to each other

Since the volume of soda remains the same when poured from the can to the glass, we can set the two volumes equal:

\[ V_{\text{can}} = V_{\text{glass}} \]

Substituting the volumes we calculated:

\[ 6\pi = 2.25\pi h \]

### Step 4: Solve for \( h \)

1. **Cancel \( \pi \) from both sides:**
\[ 6 = 2.25h \]

2. **Solve for \( h \):**
\[ h = \frac{6}{2.25} \]

3. **Convert \( 2.25 \) to a fraction:**
\[ 2.25 = \frac{9}{4} \]
Thus,
\[ h = \frac{6}{\frac{9}{4}} = 6 \times \frac{4}{9} = \frac{24}{9} = \frac{8}{3} \]

### Step 5: Convert to mixed fraction

1. **Convert \( \frac{8}{3} \) to a mixed fraction:**
- Divide 8 by 3:
\[ 8 \div 3 = 2 \quad \text{(remainder 2)} \]
- Thus, \( \frac{8}{3} = 2 \frac{2}{3} \).

### Final Answer

The height of the soda pop in the glass is \( \frac{8}{3} \) inches or \( 2 \frac{2}{3} \) inches.

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