Name: Cross Section of Solid Shapes
Uploaded: 2019-07-19
Duration: 4 min 33 s
Description: Did you know that 3D shapes are dissected from different planes, then every cross-section might look like a different plane figure? Let's take a look at this...

Question

The cross-section of a tunnel is perpendicular to its length is a trapezium $A B C D$ as shown in the following figure. Also, $A M = B N, A B = 7 m$ and $C D = 5 m$ . The height of the tunnel is $2.4 m$ . The tunnel is $40 m$ long. Calculate the cost of painting the internal surface of the tunnel(excluding the floor) at the rate of $₹ 5$ per $m^{2}$ .

Accepted Answer

Given, the cross-section of a tunnel is perpendicular to its length is a trapezium $A B C D$ .

And $A M = B N, A B = 7 m$ and $C D = 5 m$

The height of the tunnel $= 2.4 m$

Thus, $D M = C N = 2.4 m$

$∴ A M = B N = \frac{A B - M N}{2} = \frac{A B - C D}{2} = (\frac{7 - 5}{2}) m = 1 m$

Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In $△ D A M, A D^{2} = A M^{2} + D M^{2}$

$\Rightarrow A D^{2} = 1^{2} + {(2.4)}^{2}$

$\Rightarrow A D^{2} = 1 + 5.76$

$\Rightarrow A D = \sqrt{6.76}$

$\Rightarrow A D = 2.6$

Therefore, $A D = C B = 2.6 m$

Perimeter of the cross-section of the tunnel $= A B + A D + C B + C D$

$= 7 m + 2.6 m + 2.6 m + 5 m$

$= 17.2 m$

Length of the tunnel $= 40 m$

Surface area $=$ Perimeter of the cross-section $\times$ Length

Internal surface area of the tunnel $=$ $($ Length of the tunnel $\times$ Perimeter of the cross-section of the tunnel $) - ($ Length of the tunnel $\times A B)$

$= \{(17.2 \times 40) - (40 \times 7)\} m^{2}$

$= (688 - 280) m^{2}$

$= 408 m^{2}$

Cost of painting the internal surface of the tunnel $= ₹ 5$ per $m^{2}$

Total cost of painting $=$ Internal surface area of the tunnel $\times$ Rate of painting per $m^{2}$

$= ₹ (408 \times 5)$

$= ₹ 2040$

Therefore, the required answer is $₹ 2040$ .

Important Questions on Solids [Surface Area and Volume of 3-D Solids]

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