A circus tent in the form of a cylinder is surmounted by a cone. The height of the tent is $13 \; m$ and the height of the cylinder is $8 \; m$. If the diameter of its base is $24 \; m$, calculate the total surface area of the tent to the nearest square meter. Take $\pi = \dfrac{22}{7}$
Height of tent $= 13 \; m$ (given)
Height of cylinder $= H = 8 \; m$ (given)
$\therefore \;$ Height of cone $= h = 13 - 8 = 5 \; m$
Diameter of base of tent $= 24 \; m$ (given)
$\therefore \;$ Radius of cylinder $= $ Radius of cone $= R = 12 \; m$
Slant height of cone $= \ell = \sqrt{h^2 + R^2} = \sqrt{5^2 + 12^2} = \sqrt{169} = 13 \; m$
Surface area of the conical portion of the tent $= s = \pi R \left(\ell + R\right)$
i.e. $\;$ $s = \pi \times 12 \left(13 + 12\right) = \pi \times 12 \times 25 = 300 \pi \; m^2$
Surface area of the cylindrical portion of the tent $= S = 2 \pi R \left(H + R\right)$
i.e. $\;$ $S = 2 \times \pi \times 12 \left(8 + 12\right) = \pi \times 24 \times 20 = 480 \pi \; m^2$
$\therefore \;$ Total surface area of the tent $= A = s + S$
i.e. $\;$ $A = 300 \pi + 480 \pi = 780 \times \dfrac{22}{7} = 2451.4 \; m^2$
i.e. $\;$ Total surface area of the tent $= 2451 \; m^2$ (to the nearest square meter)