Question

In the figure, a tent is in the shape of a cylinder surmounted by a conical top of the same diameter. If the height and diameter of the cylindrical part are 2.1 m and 3 m respectively and the slant height of the conical part is 2.8 m, find the cost of canvas needed to make the tent if the canvas is available at the rate of Rs. 500 sq. meter. Consider $\left( {\pi = \dfrac{{22}}{7}} \right)$.

               

Answer 1

Hint: We will find the curved surface area of the cylinder and the cone and sum them up to find the required surface area of the canvas and then find the cost by multiplying it by 500.

Complete step-by-step answer:
Let us first discuss what we need to find in the question.
We have a tent in the shape of a cylinder surmounted by a cone on its top.
Since, we are not given that the base of the tent is also there, we will consider that the tent does not have a base. So, we will just find the curved surface area of the cylinder. Now, the cone is the top part, so it must not have a base as well. We will find only the curved surface area of this as well.
We know that the curved surface area of a cylinder is given by ${S_1} = 2\pi rh$ …………(1), where r is the radius of the base of the cylinder and h is the height of the cylinder. We are denoting the cylinder’s CSA by a 1 in the subscript of S.
Now, the surface area of a cone is given by ${S_2} = \pi rl$ ……….(2), where r is the radius of base and l is the slant height of the cone. We are denoting the cylinder’s CSA by a 2 in the subscript of S.
By using (1) and (2):
The total surface area of required canvas is $S = {S_1} + {S_2} = 2\pi rh + \pi rl = \pi r(2h + l)$ ……..(3)
Now, we are already given that the diameter of the cone and cylinder is 3 m.
Since, $radius = \dfrac{{diameter}}{2}$.
Therefore, $r = \dfrac{3}{2}m$.
And we are already given that height (h) and slant height (l) are 2.1 m and 2.8 m respectively.
Now, putting all these values in (3), we will have:-
 $ \Rightarrow S = \pi r(2h + l) = \dfrac{{22}}{7} \times \dfrac{3}{2} \times \{ (2 \times 2.1) + 2.8\} {m^2}$
$ \Rightarrow S = \dfrac{{11}}{7} \times 3 \times (4.2 + 2.8){m^2}$
\[ \Rightarrow S = \dfrac{{11}}{7} \times 3 \times 7 = 33{m^2}\]
Hence, the required canvas is 33 square meter.
Cost of 1 square meter = Rs. 500
Cost of 33 square meter = $Rs.(33 \times 500) = Rs.16500$.
Hence, the answer is Rs. 16500.

Note: The students must keep in mind that they are already given to use a specific value of $\pi $ in the question.
You may also note that sometimes, it may be given that the tent has a base, then you must add the surface area of the base that is the area of the circular part as well.
At no point in the question, you should forget to write the units because a number 5 does not represent anything. Neither that it is cost, not that it is an area.