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The cost of canvas required for a conical tent of height 8m and diameter of base 12m at the rate of Rs.3.50 per ${m^2}$ is

Last updated date: 20th Jun 2024
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Hint: As we are asked for the cost of the canvas required for the tent it is sufficient to find the curved surface area of the cone because we just need the outer surface area in order to make the tent so we don’t include the area of the base. We can use the formula $\pi rl$ to calculate the curved surface area of the cone, but we need the slant height but it is not given in the problem so we can calculate the slant height using the formula $l = \sqrt {{h^2} + {r^2}} $. After finding the area we need to multiply it by 3.50, as it is the cost of canvas per ${m^2}$ to obtain the total cost of the canvas required for the tent.

Complete step by step answer:

Step 1:We are given the height and base diameter of the cone

Height, h =8m
Diameter, d =12m
From the diameter,let us calculate the radius ,r
$r = \dfrac{d}{2} = \dfrac{{12}}{2} = 6m$

Step 2:The formula of the curved surface area of a cone is $\pi rl$
So we need to find the slant height , l of the cone by using the formula below
$ \Rightarrow l = \sqrt {{h^2} + {r^2}} $
By substituting the values of h and r ,
\Rightarrow l = \sqrt {{{(8)}^2} + {{(6)}^2}} \\
\Rightarrow l = \sqrt {64 + 36} \\
\Rightarrow l = \sqrt {100} \\
\Rightarrow l = 10m \\
\end{gathered} $
Step 3: Now ,we have obtained our slant height .We can find the curved surface area of the cone
Curved Surface Area of the cone (CSA) =$\pi rl$sq ${m^2}$
\Rightarrow CSA = \dfrac{{22}}{7}*6*10 \\
\Rightarrow CSA = \dfrac{{22*60}}{\begin{gathered}
  7 \\
\end{gathered} } \\
\Rightarrow CSA = \dfrac{{1320}}{7} \\
\Rightarrow CSA = 188.57{\text{ }}{m^2} \\
\end{gathered} $

Step 4: We are given that the rate of canvas per ${m^2}$ is Rs.3.50.The canvas required for the conical tent is equal to the curved surface area.
Therefore ,the cost of canvas for $188.57{m^2} = 3.50*188.57 = 659.99$
By rounding off the value we get the cost to be Rs.660
The correct option is d

Note: Since its just the canvas required to make a tent, its enough if we find a curved surface area, not the total surface area. Many students make a mistake here.
Many students forget to find the slant height and directly substitute the value of height in the place of slant height.