A cylinder is of the height 8 m and has base radius 8 m. The maximum length of the rod that can be placed in it is

Options

A. `8 sqrt(5) m `
B. `9 sqrt(5) m `
C. `8 sqrt(3)` m
D. `8 ( 2 pi + 1) m `

Answer

To find the maximum length of the rod that can be placed in a cylinder with a height of 8 m and a base radius of 8 m, we can follow these steps:

### Step 1: Identify the dimensions of the cylinder
- Height (h) = 8 m
- Radius (r) = 8 m
- Diameter (d) = 2 * r = 2 * 8 = 16 m

### Step 2: Understand the geometry
The maximum length of the rod that can be placed inside the cylinder is the length of the diagonal of the cylinder. This diagonal forms a right triangle with the height and the diameter of the base of the cylinder.

### Step 3: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ \text{Diagonal}^2 = \text{Height}^2 + \text{Diameter}^2 \]
Let the diagonal (maximum length of the rod) be denoted as \( L \).

### Step 4: Substitute the values
Substituting the known values into the equation:
\[ L^2 = h^2 + d^2 \]
\[ L^2 = 8^2 + 16^2 \]

### Step 5: Calculate the squares
Calculating the squares:
\[ L^2 = 64 + 256 \]
\[ L^2 = 320 \]

### Step 6: Take the square root
To find \( L \), take the square root of both sides:
\[ L = \sqrt{320} \]

### Step 7: Simplify the square root
We can simplify \( \sqrt{320} \):
\[ \sqrt{320} = \sqrt{64 \times 5} = \sqrt{64} \times \sqrt{5} = 8\sqrt{5} \]

### Final Answer
Thus, the maximum length of the rod that can be placed in the cylinder is:
\[ \boxed{8\sqrt{5} \text{ m}} \]