An association offered to contribute 50% of the cost for making 100 tents. The lower part of each tent is of the form of a cylinder of diameter 4.2 m and height 4 m with the conical upper part of same diameter but of height 2.8 m. If the canvas to be used costs ₹ 100 per square meter, find the amount the association pays.
Diameter $= 4.2 \ m$
$\therefore$ Radius $= r = 2.1 \ m$
Height $=h=2.8 \ m$
$\begin{aligned} \therefore \text{Slant height} = \ell & = \sqrt{h^2+r^2} \\\\ & = \sqrt{2.8^2 + 2.1^2} \\\\ & = \sqrt{12.25} = 3.5 \ m \end{aligned}$
$\therefore$ Curved surface area of the conical portion
$= \pi r \ell = \dfrac{22}{7} \times 2.1 \times 3.5 = 23.1 \ m^2$ $\cdots$ (1)
For the cylindrical portion of the tent:
Diameter $= 4.2 \ m$
$\therefore$ Radius $= R = 2.1 \ m$
Height $=H = 4 \ m$
$\therefore$ Curved surface area of the cylindrical portion $=2\pi R H = 2 \times \dfrac{22}{7} \times 2.1 \times 4 = 52.8 \ m^2$ $\cdots$ (2)
For the entire tent:
$\therefore$ From equations (1) and (2),
curved surface area of the entire tent $=23.1+52.8 = 75.9 \ m^2$
Cost of canvas per square meter $=$ ₹ 100
$\therefore$ Cost of canvas for 1 tent $= 75.9 \times 100 =$ ₹ 7,590
$\therefore$ Cost of canvas for 100 tents $=7590 \times 100 = $ ₹ 7,59,000
$\therefore$ Amount paid by the association $= 50\%$ of ₹ 7,59,000 $= \dfrac{759000}{2}=$ ₹ 3,79,500