Let's break down the problem step-by-step to find the value of [tex]\( k \)[/tex].
1. Determine the dimensions of the cuboid and its volume:
The cuboid formed by the concrete blocks has dimensions [tex]\( x \times x \times 2x \)[/tex].
The volume [tex]\( V_{\text{cuboid}} \)[/tex] of a cuboid is given by:
[tex]\[ V_{\text{cuboid}} = \text{length} \times \text{width} \times \text{height} \][/tex]
For our cuboid:
[tex]\[ V_{\text{cuboid}} = x \times x \times 2x = 2x^3 \][/tex]
2. Determine the dimensions of the cylindrical tunnel and its volume:
The tunnel has a cylindrical shape with a radius [tex]\( \frac{x}{2} \)[/tex] and height [tex]\( 2x \)[/tex].
The volume [tex]\( V_{\text{tunnel}} \)[/tex] of a cylinder is given by:
[tex]\[ V_{\text{tunnel}} = \pi \times (\text{radius})^2 \times \text{height} \][/tex]
For our tunnel:
[tex]\[ \text{radius} = \frac{x}{2} \][/tex]
[tex]\[ \text{height} = 2x \][/tex]
Therefore, the volume [tex]\( V_{\text{tunnel}} \)[/tex] is:
[tex]\[ V_{\text{tunnel}} = \pi \left( \frac{x}{2} \right)^2 \times 2x \][/tex]
[tex]\[ V_{\text{tunnel}} = \pi \left( \frac{x^2}{4} \right) \times 2x \][/tex]
[tex]\[ V_{\text{tunnel}} = \pi \left( \frac{x^2}{2} \right) \times x \][/tex]
[tex]\[ V_{\text{tunnel}} = \pi \times \frac{x^3}{2} \][/tex]
[tex]\[ V_{\text{tunnel}} = \frac{\pi x^3}{2} \][/tex]
3. Find the ratio of the volume of the tunnel to the volume of the cuboid:
The ratio is given by:
[tex]\[ \frac{V_{\text{tunnel}}}{V_{\text{cuboid}}} = \frac{\frac{\pi x^3}{2}}{2x^3} \][/tex]
[tex]\[ \frac{V_{\text{tunnel}}}{V_{\text{cuboid}}} = \frac{\pi x^3}{2} \div 2x^3 \][/tex]
[tex]\[ \frac{V_{\text{tunnel}}}{V_{\text{cuboid}}} = \frac{\pi x^3}{2 \cdot 2x^3} \][/tex]
[tex]\[ \frac{V_{\text{tunnel}}}{V_{\text{cuboid}}} = \frac{\pi}{4} \][/tex]
Given the ratio:
[tex]\[ \frac{V_{\text{tunnel}}}{V_{\text{cuboid}}} = \frac{\pi}{k} \][/tex]
We have already calculated:
[tex]\[ \frac{V_{\text{tunnel}}}{V_{\text{cuboid}}} = \frac{\pi}{4} \][/tex]
Therefore, we equate:
[tex]\[ \frac{\pi}{4} = \frac{\pi}{k} \][/tex]
From this, it follows that:
[tex]\[ k = 4 \][/tex]
Thus, the value of [tex]\( k \)[/tex] is [tex]\( 1.27323954473516 \)[/tex].
