A slab of ice 8 inches in length, 11 inches in breadth and 2 inches thick was melted and resolidified in the form of a rod of 8 inches diameter. What is the length of such a rod in inches ?

Answer

To find the length of the rod formed from the melted slab of ice, we need to follow these steps:

### Step 1: Calculate the volume of the slab of ice.
The volume \( V \) of a rectangular prism (the slab of ice) is given by the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
Here, the dimensions of the slab of ice are:
- Length = 8 inches
- Breadth = 11 inches
- Height (thickness) = 2 inches

Substituting the values:
\[ V = 8 \times 11 \times 2 \]
\[ V = 176 \text{ cubic inches} \]

### Step 2: Calculate the volume of the rod.
The volume \( V \) of a cylinder (the rod) is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height (length) of the cylinder.

The diameter of the rod is given as 8 inches, so the radius \( r \) is:
\[ r = \frac{\text{diameter}}{2} = \frac{8}{2} = 4 \text{ inches} \]

Now, we can express the volume of the rod in terms of its height \( h \):
\[ V = \pi (4)^2 h = 16\pi h \]

### Step 3: Set the volumes equal to each other.
Since the volume of the melted ice slab is equal to the volume of the rod:
\[ 176 = 16\pi h \]

### Step 4: Solve for \( h \).
To find \( h \), we rearrange the equation:
\[ h = \frac{176}{16\pi} \]

Calculating \( h \):
\[ h = \frac{176}{16 \times 3.14} \quad (\text{using } \pi \approx 3.14) \]
\[ h = \frac{176}{50.24} \approx 3.5 \text{ inches} \]

### Final Answer:
The length of the rod is approximately **3.5 inches**.

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