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A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40cm and height 1m. If the outer side of each of the cones is to be painted and the cost of painting is Rs.12 per ${m^2}$, what will be the cost of painting all these cones? (Use $\pi = 3.14$ and take $\sqrt {1.04} = 1.02$ ).

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Last updated date: 15th Jun 2024
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Answer
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Hint: We are given that 50 cones are needed to be painted and we are given the base diameter and height of the cone. Since they are of different units we convert them to a common unit and then to find the CSA we need slant height which is given by the formula $l = \sqrt {{h^2} + {r^2}} $and we can find the CSA using the formula $CSA = \pi rl$. And we get the Area of one cone and multiplying it by 12 we get the cost of painting 1 cone and further multiplying by 50 we get the cost of painting 50 cones.

Complete step by step solution:
We are given that 50 hollow cones are used and 50 cones need to be painted.
Since only the outer surface is needed to be painted it is enough if we find the curved surface area as there is no need to add the base area of the cone.
We are given that the cone has a base diameter of 40 cm and height 1 m.
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Since the diameter and height are of different units we need to convert them to one common unit
Since the rate of painting is given in ${m^2}$ lets convert them to metres
To convert centimetre to metre we need to divide it by 100
$ \Rightarrow \dfrac{{40}}{{100}} = 0.4m$
Hence now our base diameter is 0.4 m
We get our radius by dividing it by 2
$ \Rightarrow r = \dfrac{{0.4}}{2} = 0.2m$
The formula for the curved surface area of a cone is $\pi rl$
Where r is the radius and l is the slant height
Hence we need to find our slant height by using the formula
$ \Rightarrow l = \sqrt {{h^2} + {r^2}} $
Using the known values we get
\[
   \Rightarrow l = \sqrt {{{\left( {0.2} \right)}^2} + {1^2}} \\
   \Rightarrow l = \sqrt {0.04 + 1} \\
   \Rightarrow l = \sqrt {1.04} = 1.02 \\
 \]
Now we have our slant height
Hence ,
$
   \Rightarrow CSA = \pi rl \\
   \Rightarrow CSA = 3.14\times 0.2\times 1.02 \\
   \Rightarrow CSA = 0.64056{m^2} \\
 $
Hence area needed to be painted for one cone is $0.64056{m^2}$
We are given that it costs Rs.12 per${m^2}$
The cost of painting one cone is given by
$ \Rightarrow 0.64056 \times 12 = Rs.7.68672$
Therefore the cost of painting 50 cones is given as
$ \Rightarrow 7.68672 \times 50 = 384.336$

Hence the cost of painting 50 cones is Rs.384.336

Note :
Here many students get confused as we are given that it is a hollow cone.
Since we are asked only for the curved surface area it doesn’t matter whether it is a hollow cone or a solid cone.
And students also use the height of the cone instead of the slant height which is wrong.