Question

A series of three? separate, adjacent tunnels is constructed through a mountain. Its length is approximately 25 kilometers. Each of the three tunnels is shaped like a? half-cylinder with a radius of 5 meters? (the height of the? tunnel) and a length of 25 kilometers? (the length of the? tunnel). How much earth? (volume) was removed to build the three? tunnels

Answer 1

`312.5\pi \text{ km}^3\approx 981.75\text{ km}^3`

Explanation:

We have been given that a series of 3 separate, adjacent tunnels is constructed through a mountain. Its length is approximately 25 kilometers.

Each of the three tunnels is shaped like a half-cylinder with a radius of 5 meters.

Since we know that volume of a semicircular or a half cylinder is half the volume of a circular cylinder.

`\text{Volume of a semicircular cylinder}=(\pi r^2h)/(2)`, where,

r = Radius of cylinder,

h = height of the cylinder.

Upon substituting our given values in volume formula we will get,

`\text{Volume of a semicircular cylinder}=\frac{\pi (5\text{ km})^2*25\text{ km}}{2}`

`\text{Volume of a semicircular cylinder}=\frac{\pi*25\text{ km}^2*25\text{ km}}{2}`

`\text{Volume of a semicircular cylinder}=\frac{\pi*625\text{ km}^3}{2}`

`\text{Volume of a semicircular cylinder}=\pi*312.5\text{ km}^3`

`\text{Volume of a semicircular cylinder}=\pi*312.5\text{ km}^3`

`\text{Volume of a semicircular cylinder}=981.74770\text{ km}^3`

Therefore, the volume of earth removed to build the three tunnels is
`312.5\pi \text{ km}^3\approx 981.75\text{ km}^3`.