A military tent of height 8.25m is in the form of a right circular cylinder of base diameter 30m and height 5.5m surmounted by a right circular cone of same base radius. Find the length of the canvas used in making the tent if the breadth of the canvas is 1.5m.
VerifiedHint: Use the formula: Height of the military tent = Height of the circular cylinder + Height of the right circular cone to find the height of the cone. Then use $l=\sqrt{{{r}^{2}}+{{h}^{2}}}$ to find the slant height of the cone. Then use the formulae surface are of cone = $\pi rl$, and surface area of cylinder = $2\pi rh$. Add these to get the total area and divide this total area by the given breadth to get the final answer.
Complete step by step solution:
In this question we are given that a military tent of height 8.25m is in the form of a right circular cylinder of base diameter 30m and height 5.5m surmounted by a right circular cone of same base radius.
We need to find the length of the canvas used in making the tent if the breadth of the canvas is 1.5m.
Height of the military tent = Height of the circular cylinder + Height of the right circular cone
Given,
Height of the military tent = 8.25m
Height of the circular cylinder = 5.5m
Therefore, height of the right circular cone = 8.25 − 5.5m = 2.75m
We know that,
The slant height of a cone, l is equal to the root of the sum of squares of its radius and its height.
i.e. $l=\sqrt{{{r}^{2}}+{{h}^{2}}}$
where, r= radius of base and h= altitude height of cone
We are given that, radius of cylinder = radius of cone
So, radius is half of diameter which is 30m
So, the radius r = 15m.
$l=\sqrt{{{15}^{2}}+{{2.75}^{2}}}$
$l=\sqrt{225+7.5625}=15.25$
So, the slant height of the cone, l = 15.25 m.
Now, surface area of cone = $\pi rl=\pi \times 15\times 15.25=718.9{{m}^{2}}$
Surface area of cylinder = $2\pi rh=2\times \pi \times 15\times 5.5=518.57{{m}^{2}}$
Total area of the canvas = Surface area of cone + Surface area of cylinder = 718.9 + 518.57
Total area of the canvas = 1237.47 ${{m}^{2}}$
Now, the breadth of canvas is given to be 1.5 m. We will find the length by dividing the total area by the breadth.
So, length of the canvas = $\dfrac{1237.47}{1.5}=824.98m$
Hence, the length of the canvas used is 824.98m.
Note: In this question, it is very important to note that we are using the curved surface area for the cone. Similarly, we are using the curved surface area of the cylinder and not the total surface areas of these solid figures. Some students might use total surface area which will give the wrong answer. So always think properly which formulae to use.
