The cross-section of a tunnel is a square of side 7 m surmounted by a semicircle as shown in the following figure.The tunnel is 80 m long. Calculate: (i) its volume (ii) the surface area of the tunnel (excluding the floor)
The cross-section of a tunnel is a square of side 7 m surmounted by a semicircle as shown in the following figure.The tunnel is 80 m long. Calculate: (i) its volume (ii) the surface area of the tunnel (excluding the floor)

FIGURE:

Selina Solutions Concise Class 10 Maths Chapter 20 ex. 20(G) - 3

Side of square (a) = 7m

the radius of semi-circle = 7/2 m

Length of the tunnel = 80 m

    \[\begin{array}{*{35}{l}} Area\text{ }of\text{ }cross\text{ }section\text{ }of\text{ }the\text{ }front\text{ }part\text{ }=\text{ }{{a}^{2}}~+\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }\pi {{r}^{2}}  \\ =\text{ }7\text{ }x\text{ }7\text{ }+\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }x\text{ }22/7\text{ }x\text{ }7/2\text{ }x\text{ }7/2  \\ =\text{ }49\text{ }+\text{ }77/4\text{ }{{m}^{2}}  \\ =\text{ }\left( 196\text{ }+\text{ }77 \right)/\text{ }4  \\ =\text{ }273/4\text{ }{{m}^{2}}  \\ \end{array}\]

(i) Thus, the volume of the turnel = area of cross section x length of the turnel

    \[\begin{array}{*{35}{l}} =\text{ }273/4\text{ }x\text{ }80  \\ =\text{ }5460\text{ }{{m}^{3}}  \\ \end{array}\]

(ii) Circumference of the front of tunnel (excluding the floor) = 2a + ½ x 2πr

    \[\begin{array}{*{35}{l}} =\text{ }2\text{ }x\text{ }7\text{ }+\text{ }22/7\text{ }x\text{ }7/2  \\ =\text{ }14\text{ }+\text{ }11\text{ }=\text{ }25\text{ }cm  \\ \end{array}\]

Hence, the surface area of the inner part of the tunnel = 25 x 80 = 2000 m2