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. `8sqrt(5)m`
B. `9sqrt(5)m`
C. `8sqrt(3)cm`
D. `8(2pi+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 visualize the situation and apply the Pythagorean theorem.

### Step-by-Step Solution:

1. **Identify the dimensions of the cylinder**:
- Height (h) = 8 m
- Radius (r) = 8 m

2. **Visualize the rod inside the cylinder**:
- The rod will be placed diagonally from one point on the base of the cylinder to the opposite point on the top edge of the cylinder. This forms a right triangle.

3. **Determine the triangle's sides**:
- The height of the cylinder (h) will be one side of the triangle (perpendicular).
- The diameter of the base of the cylinder will be the other side of the triangle (base). The diameter (d) is twice the radius, so:
\[ d = 2 \times r = 2 \times 8 = 16 \text{ m} \]

4. **Apply the Pythagorean theorem**:
- In a right triangle, the square of the hypotenuse (length of the rod, AC) is equal to the sum of the squares of the other two sides:
\[ AC^2 = AB^2 + BC^2 \]
- Here, \( AB \) (height) = 8 m and \( BC \) (diameter) = 16 m.
- Substitute the values:
\[ AC^2 = 8^2 + 16^2 \]
\[ AC^2 = 64 + 256 \]
\[ AC^2 = 320 \]

5. **Calculate the length of the rod**:
- To find \( AC \), take the square root of 320:
\[ AC = \sqrt{320} \]
- Simplifying \( \sqrt{320} \):
\[ \sqrt{320} = \sqrt{64 \times 5} = 8\sqrt{5} \]
- Approximating \( \sqrt{5} \) (approximately 2.236):
\[ AC \approx 8 \times 2.236 \approx 17.888 \text{ m} \]

Thus, the maximum length of the rod that can be placed in the cylinder is approximately **17.89 m**.