Question

calculate the volume of the shell of this this tunnel.
give your answer in appropriate units, and round your answer to 1 decimal place.

Explanation:

Identify the geometric shape

The tunnel shell is a half of a hollow cylinder (a semicircular cylindrical shell).

• Outer radius \(R = 7\text{ m}\)
• Thickness \(t = 1.5\text{ m}\)
• Inner radius \(r = R - t = 7 - 1.5 = 5.5\text{ m}\)
• Length \(L = 20\text{ m}\)

Calculate the cross-sectional area

Using the Hollow Cylinder Volume concept, we first find the area of the semicircular ring cross-section:

$$ A = \frac{1}{2} \pi (R^2 - r^2) $$

$$ A = \frac{1}{2} \pi (7^2 - 5.5^2) = \frac{1}{2} \pi (49 - 30.25) = \frac{1}{2} \pi (18.75) = 9.375\pi\text{ m}^2 $$

Calculate the volume of the shell

Multiply the cross-sectional area by the length of the tunnel:

$$ V = A \times L = 9.375\pi \times 20 = 187.5\pi\text{ m}^3 $$

Round to one decimal place

Using \(\pi \approx 3.14159265\):

$$ V \approx 187.5 \times 3.14159265 \approx 589.0486\text{ m}^3 $$

Rounding to 1 decimal place gives \(589.0\text{ m}^3\).

Answer:

\(589.0\text{ m}^3\)