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Question

A military tent of height 8.25 m is in the form of a circular cylinder of base diameter 30 m and height 5.5 m surmounted by a right circular cone of same base radius.Find the length of canvas used in making the tent,if the breath of the canvas is 1.5 m.

Solution
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A military tent is made as a combination of right circular and right circular cone on top.
Given:
Total height of tent = h = 8.25 m
Base diameter of tent = 30 m, then
Base radius of tent = r = 30/2 m = 15 m
Height of right circular cylinder = 5.5 m
Base radius of cone = 15 m
Base radius of cone = 15 m
Let slant height of cone = I m
Now,
Curved surface area of right circular part of tent = $$ 2 \pi rh $$
and
Height of conical part = total height of tent - height of cyclindrical part
height of cone = 8.25 = 5.5 = 2.75 m
$$l^2 = h^2 + r^2$$
$$ = 2.75^2 + 15^2$$
= 232.5625
or I = 15.25 m
Curved surface area of conical part of the tent = $$ \pi rl $$
Total surface area of the tent = Curved surface area of cylindrical part + curved surface area of conical part
Total surface area of tent = $$ 2 \pi rh + \pi rl $$
$$ = \pi r (2h+ l) $$
= $$ \dfrac{22}{7} \times 15 \times (2 \times 5+15.25) $$
= 1237.5
Total surface area of tent is $$1237.5 m^2$$
Length of canvas used = 1.5 m
Length of canvas used x breadth of canvas used = Total surface area of tent
length of canvas used = 1237.5 /1.5 = 825
Length of canvas used is 825 m.

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